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Metamath Proof Explorer

Theorem marrepeval

Description: An entry of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019)

Ref Expression
Hypotheses marrepfval.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
marrepfval.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
marrepfval.q ⊢ 𝑄 = ( 𝑁 matRRep 𝑅 )
marrepfval.z ⊢ 0 = ( 0g ‘ 𝑅 )
Assertion marrepeval ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( 𝐼 ( 𝐾 ( 𝑀 𝑄 𝑆 ) 𝐿 ) 𝐽 ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 𝑆 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) )

Proof

Step Hyp Ref Expression
1 marrepfval.a ⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2 marrepfval.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
3 marrepfval.q ⊢ 𝑄 = ( 𝑁 matRRep 𝑅 )
4 marrepfval.z ⊢ 0 = ( 0g ‘ 𝑅 )
5 1 2 3 4 marrepval ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ) → ( 𝐾 ( 𝑀 𝑄 𝑆 ) 𝐿 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
6 5 3adant3 ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( 𝐾 ( 𝑀 𝑄 𝑆 ) 𝐿 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) )
7 simp3l ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → 𝐼 ∈ 𝑁 )
8 simpl3r ⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) ∧ 𝑖 = 𝐼 ) → 𝐽 ∈ 𝑁 )
9 4 fvexi ⊢ 0 ∈ V
10 ifexg ⊢ ( ( 𝑆 ∈ ( Base ‘ 𝑅 ) ∧ 0 ∈ V ) → if ( 𝑗 = 𝐿 , 𝑆 , 0 ) ∈ V )
11 9 10 mpan2 ⊢ ( 𝑆 ∈ ( Base ‘ 𝑅 ) → if ( 𝑗 = 𝐿 , 𝑆 , 0 ) ∈ V )
12 ovexd ⊢ ( 𝑆 ∈ ( Base ‘ 𝑅 ) → ( 𝑖 𝑀 𝑗 ) ∈ V )
13 11 12 ifcld ⊢ ( 𝑆 ∈ ( Base ‘ 𝑅 ) → if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ∈ V )
14 13 adantl ⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) → if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ∈ V )
15 14 3ad2ant1 ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ∈ V )
16 15 adantr ⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) ∧ ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) ) → if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ∈ V )
17 eqeq1 ⊢ ( 𝑖 = 𝐼 → ( 𝑖 = 𝐾 ↔ 𝐼 = 𝐾 ) )
18 17 adantr ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) → ( 𝑖 = 𝐾 ↔ 𝐼 = 𝐾 ) )
19 eqeq1 ⊢ ( 𝑗 = 𝐽 → ( 𝑗 = 𝐿 ↔ 𝐽 = 𝐿 ) )
20 19 ifbid ⊢ ( 𝑗 = 𝐽 → if ( 𝑗 = 𝐿 , 𝑆 , 0 ) = if ( 𝐽 = 𝐿 , 𝑆 , 0 ) )
21 20 adantl ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) → if ( 𝑗 = 𝐿 , 𝑆 , 0 ) = if ( 𝐽 = 𝐿 , 𝑆 , 0 ) )
22 oveq12 ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) → ( 𝑖 𝑀 𝑗 ) = ( 𝐼 𝑀 𝐽 ) )
23 18 21 22 ifbieq12d ⊢ ( ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) → if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 𝑆 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) )
24 23 adantl ⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) ∧ ( 𝑖 = 𝐼 ∧ 𝑗 = 𝐽 ) ) → if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 𝑆 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) )
25 7 8 16 24 ovmpodv2 ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( ( 𝐾 ( 𝑀 𝑄 𝑆 ) 𝐿 ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 𝑆 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) → ( 𝐼 ( 𝐾 ( 𝑀 𝑄 𝑆 ) 𝐿 ) 𝐽 ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 𝑆 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) ) )
26 6 25 mpd ⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ ( 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ) ) → ( 𝐼 ( 𝐾 ( 𝑀 𝑄 𝑆 ) 𝐿 ) 𝐽 ) = if ( 𝐼 = 𝐾 , if ( 𝐽 = 𝐿 , 𝑆 , 0 ) , ( 𝐼 𝑀 𝐽 ) ) )