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 = ( ∥r𝑅 )
dvdsr0.z 0 = ( 0g𝑅 )
Assertion dvdsr02 ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 0 𝑋𝑋 = 0 ) )

Proof

Step Hyp Ref Expression
1 dvdsr0.b 𝐵 = ( Base ‘ 𝑅 )
2 dvdsr0.d = ( ∥r𝑅 )
3 dvdsr0.z 0 = ( 0g𝑅 )
4 1 3 ring0cl ( 𝑅 ∈ Ring → 0𝐵 )
5 4 adantr ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → 0𝐵 )
6 eqid ( .r𝑅 ) = ( .r𝑅 )
7 1 2 6 dvdsr2 ( 0𝐵 → ( 0 𝑋 ↔ ∃ 𝑥𝐵 ( 𝑥 ( .r𝑅 ) 0 ) = 𝑋 ) )
8 5 7 syl ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 0 𝑋 ↔ ∃ 𝑥𝐵 ( 𝑥 ( .r𝑅 ) 0 ) = 𝑋 ) )
9 1 6 3 ringrz ( ( 𝑅 ∈ Ring ∧ 𝑥𝐵 ) → ( 𝑥 ( .r𝑅 ) 0 ) = 0 )
10 9 eqeq1d ( ( 𝑅 ∈ Ring ∧ 𝑥𝐵 ) → ( ( 𝑥 ( .r𝑅 ) 0 ) = 𝑋0 = 𝑋 ) )
11 eqcom ( 0 = 𝑋𝑋 = 0 )
12 10 11 bitrdi ( ( 𝑅 ∈ Ring ∧ 𝑥𝐵 ) → ( ( 𝑥 ( .r𝑅 ) 0 ) = 𝑋𝑋 = 0 ) )
13 12 rexbidva ( 𝑅 ∈ Ring → ( ∃ 𝑥𝐵 ( 𝑥 ( .r𝑅 ) 0 ) = 𝑋 ↔ ∃ 𝑥𝐵 𝑋 = 0 ) )
14 ringgrp ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
15 1 grpbn0 ( 𝑅 ∈ Grp → 𝐵 ≠ ∅ )
16 r19.9rzv ( 𝐵 ≠ ∅ → ( 𝑋 = 0 ↔ ∃ 𝑥𝐵 𝑋 = 0 ) )
17 14 15 16 3syl ( 𝑅 ∈ Ring → ( 𝑋 = 0 ↔ ∃ 𝑥𝐵 𝑋 = 0 ) )
18 13 17 bitr4d ( 𝑅 ∈ Ring → ( ∃ 𝑥𝐵 ( 𝑥 ( .r𝑅 ) 0 ) = 𝑋𝑋 = 0 ) )
19 18 adantr ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( ∃ 𝑥𝐵 ( 𝑥 ( .r𝑅 ) 0 ) = 𝑋𝑋 = 0 ) )
20 8 19 bitrd ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 0 𝑋𝑋 = 0 ) )