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Metamath Proof Explorer

Theorem dvdsr02

Description: Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015)

Ref Expression
Hypotheses dvdsr0.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
dvdsr0.d ⊢ ∥ = ( ∥r ‘ 𝑅 )
dvdsr0.z ⊢ 0 = ( 0g ‘ 𝑅 )
Assertion dvdsr02 ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 0 ∥ 𝑋 ↔ 𝑋 = 0 ) )

Proof

Step Hyp Ref Expression
1 dvdsr0.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 dvdsr0.d ⊢ ∥ = ( ∥r ‘ 𝑅 )
3 dvdsr0.z ⊢ 0 = ( 0g ‘ 𝑅 )
4 1 3 ring0cl ⊢ ( 𝑅 ∈ Ring → 0 ∈ 𝐵 )
5 4 adantr ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → 0 ∈ 𝐵 )
6 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
7 1 2 6 dvdsr2 ⊢ ( 0 ∈ 𝐵 → ( 0 ∥ 𝑋 ↔ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 0 ) = 𝑋 ) )
8 5 7 syl ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 0 ∥ 𝑋 ↔ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 0 ) = 𝑋 ) )
9 1 6 3 ringrz ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ( .r ‘ 𝑅 ) 0 ) = 0 )
10 9 eqeq1d ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 0 ) = 𝑋 ↔ 0 = 𝑋 ) )
11 eqcom ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 )
12 10 11 bitrdi ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 0 ) = 𝑋 ↔ 𝑋 = 0 ) )
13 12 rexbidva ⊢ ( 𝑅 ∈ Ring → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 0 ) = 𝑋 ↔ ∃ 𝑥 ∈ 𝐵 𝑋 = 0 ) )
14 ringgrp ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
15 1 grpbn0 ⊢ ( 𝑅 ∈ Grp → 𝐵 ≠ ∅ )
16 r19.9rzv ⊢ ( 𝐵 ≠ ∅ → ( 𝑋 = 0 ↔ ∃ 𝑥 ∈ 𝐵 𝑋 = 0 ) )
17 14 15 16 3syl ⊢ ( 𝑅 ∈ Ring → ( 𝑋 = 0 ↔ ∃ 𝑥 ∈ 𝐵 𝑋 = 0 ) )
18 13 17 bitr4d ⊢ ( 𝑅 ∈ Ring → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 0 ) = 𝑋 ↔ 𝑋 = 0 ) )
19 18 adantr ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 0 ) = 𝑋 ↔ 𝑋 = 0 ) )
20 8 19 bitrd ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 0 ∥ 𝑋 ↔ 𝑋 = 0 ) )