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Metamath Proof Explorer

Theorem zmulcom

Description: Multiplication is commutative for integers. Proven without ax-mulcom . From this result and grpcominv1 , we can show that rationals commute under multiplication without using ax-mulcom . (Contributed by SN, 25-Jan-2025)

Ref Expression
Assertion zmulcom 
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A x. B ) = ( B x. A ) )

Proof

Step Hyp Ref Expression
1 reelznn0nn 
 |-  ( A e. ZZ <-> ( A e. NN0 \/ ( A e. RR /\ ( 0 -R A ) e. NN ) ) )
2 reelznn0nn 
 |-  ( B e. ZZ <-> ( B e. NN0 \/ ( B e. RR /\ ( 0 -R B ) e. NN ) ) )
3 nn0mulcom 
 |-  ( ( A e. NN0 /\ B e. NN0 ) -> ( A x. B ) = ( B x. A ) )
4 zmulcomlem 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ B e. NN0 ) -> ( A x. B ) = ( B x. A ) )
5 zmulcomlem 
 |-  ( ( ( B e. RR /\ ( 0 -R B ) e. NN ) /\ A e. NN0 ) -> ( B x. A ) = ( A x. B ) )
6 5 eqcomd 
 |-  ( ( ( B e. RR /\ ( 0 -R B ) e. NN ) /\ A e. NN0 ) -> ( A x. B ) = ( B x. A ) )
7 6 ancoms 
 |-  ( ( A e. NN0 /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( A x. B ) = ( B x. A ) )
8 nnmulcom 
 |-  ( ( ( 0 -R A ) e. NN /\ ( 0 -R B ) e. NN ) -> ( ( 0 -R A ) x. ( 0 -R B ) ) = ( ( 0 -R B ) x. ( 0 -R A ) ) )
9 8 ad2ant2l 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R A ) x. ( 0 -R B ) ) = ( ( 0 -R B ) x. ( 0 -R A ) ) )
10 9 oveq2d 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( ( 0 -R A ) x. ( 0 -R B ) ) ) = ( 0 -R ( ( 0 -R B ) x. ( 0 -R A ) ) ) )
11 rernegcl 
 |-  ( A e. RR -> ( 0 -R A ) e. RR )
12 11 ad2antrr 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R A ) e. RR )
13 simprr 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R B ) e. NN )
14 12 13 renegmulnnass 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R B ) ) = ( 0 -R ( ( 0 -R A ) x. ( 0 -R B ) ) ) )
15 rernegcl 
 |-  ( B e. RR -> ( 0 -R B ) e. RR )
16 15 ad2antrl 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R B ) e. RR )
17 simplr 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R A ) e. NN )
18 16 17 renegmulnnass 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R A ) ) = ( 0 -R ( ( 0 -R B ) x. ( 0 -R A ) ) ) )
19 10 14 18 3eqtr4d 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R B ) ) = ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R A ) ) )
20 19 oveq2d 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R B ) ) ) = ( 0 -R ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R A ) ) ) )
21 rernegcl 
 |-  ( ( 0 -R A ) e. RR -> ( 0 -R ( 0 -R A ) ) e. RR )
22 11 21 syl 
 |-  ( A e. RR -> ( 0 -R ( 0 -R A ) ) e. RR )
23 22 ad2antrr 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( 0 -R A ) ) e. RR )
24 23 16 remulneg2d 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R ( 0 -R B ) ) ) = ( 0 -R ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R B ) ) ) )
25 rernegcl 
 |-  ( ( 0 -R B ) e. RR -> ( 0 -R ( 0 -R B ) ) e. RR )
26 15 25 syl 
 |-  ( B e. RR -> ( 0 -R ( 0 -R B ) ) e. RR )
27 26 ad2antrl 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( 0 -R B ) ) e. RR )
28 27 12 remulneg2d 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R ( 0 -R A ) ) ) = ( 0 -R ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R A ) ) ) )
29 20 24 28 3eqtr4d 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R ( 0 -R B ) ) ) = ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R ( 0 -R A ) ) ) )
30 renegneg 
 |-  ( A e. RR -> ( 0 -R ( 0 -R A ) ) = A )
31 30 ad2antrr 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( 0 -R A ) ) = A )
32 renegneg 
 |-  ( B e. RR -> ( 0 -R ( 0 -R B ) ) = B )
33 32 ad2antrl 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( 0 -R ( 0 -R B ) ) = B )
34 31 33 oveq12d 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R A ) ) x. ( 0 -R ( 0 -R B ) ) ) = ( A x. B ) )
35 33 31 oveq12d 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( ( 0 -R ( 0 -R B ) ) x. ( 0 -R ( 0 -R A ) ) ) = ( B x. A ) )
36 29 34 35 3eqtr3d 
 |-  ( ( ( A e. RR /\ ( 0 -R A ) e. NN ) /\ ( B e. RR /\ ( 0 -R B ) e. NN ) ) -> ( A x. B ) = ( B x. A ) )
37 3 4 7 36 ccase 
 |-  ( ( ( A e. NN0 \/ ( A e. RR /\ ( 0 -R A ) e. NN ) ) /\ ( B e. NN0 \/ ( B e. RR /\ ( 0 -R B ) e. NN ) ) ) -> ( A x. B ) = ( B x. A ) )
38 1 2 37 syl2anb 
 |-  ( ( A e. ZZ /\ B e. ZZ ) -> ( A x. B ) = ( B x. A ) )