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Theorem zerdivemp1x 32712
Description: In a unitary ring a left invertible element is not a zero divisor. See also ringinvnzdiv 18362. (Contributed by Jeff Madsen, 18-Apr-2010.)
Hypotheses
Ref Expression
zerdivempx.1 𝐺 = (1st𝑅)
zerdivempx.2 𝐻 = (2nd𝑅)
zerdivempx.3 𝑍 = (GId‘𝐺)
zerdivempx.4 𝑋 = ran 𝐺
zerdivempx.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
zerdivemp1x ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ ∃𝑎𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍)))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎   𝐻,𝑎   𝑅,𝑎   𝑋,𝑎   𝑍,𝑎
Allowed substitution hints:   𝑈(𝑎)   𝐺(𝑎)

Proof of Theorem zerdivemp1x
StepHypRef Expression
1 oveq2 6535 . . . . . . 7 ((𝐴𝐻𝐵) = 𝑍 → (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍))
2 simpl1 1056 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝑅 ∈ RingOps)
3 simpr1 1059 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝑎𝑋)
4 simpr3 1061 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝐴𝑋)
5 simpl3 1058 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝐵𝑋)
6 zerdivempx.1 . . . . . . . . . . 11 𝐺 = (1st𝑅)
7 zerdivempx.2 . . . . . . . . . . 11 𝐻 = (2nd𝑅)
8 zerdivempx.4 . . . . . . . . . . 11 𝑋 = ran 𝐺
96, 7, 8rngoass 32671 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ (𝑎𝑋𝐴𝑋𝐵𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
102, 3, 4, 5, 9syl13anc 1319 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
11 eqtr 2628 . . . . . . . . . . . . 13 ((((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍))
1211ex 448 . . . . . . . . . . . 12 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)))
13 oveq1 6534 . . . . . . . . . . . . . . . 16 ((𝑎𝐻𝐴) = 𝑈 → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵))
14 eqtr 2628 . . . . . . . . . . . . . . . . . . . 20 (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
15 zerdivempx.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑍 = (GId‘𝐺)
1615, 8, 6, 7rngorz 32688 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ RingOps ∧ 𝑎𝑋) → (𝑎𝐻𝑍) = 𝑍)
17163adant3 1073 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝑎𝐻𝑍) = 𝑍)
186rneqi 5260 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ran 𝐺 = ran (1st𝑅)
198, 18eqtri 2631 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑋 = ran (1st𝑅)
20 zerdivempx.5 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑈 = (GId‘𝐻)
217, 19, 20rngolidm 32702 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑅 ∈ RingOps ∧ 𝐵𝑋) → (𝑈𝐻𝐵) = 𝐵)
22213adant2 1072 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝑈𝐻𝐵) = 𝐵)
23 simp1 1053 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
24 simp2 1054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = 𝐵)
25 simp3 1055 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑎𝐻𝑍) = 𝑍)
2623, 24, 253eqtr3d 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → 𝐵 = 𝑍)
2726a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝐴𝑋𝐵 = 𝑍))
28273exp 1255 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → ((𝑈𝐻𝐵) = 𝐵 → ((𝑎𝐻𝑍) = 𝑍 → (𝐴𝑋𝐵 = 𝑍))))
2928com14 93 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑋 → ((𝑈𝐻𝐵) = 𝐵 → ((𝑎𝐻𝑍) = 𝑍 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍))))
3029com13 85 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐻𝑍) = 𝑍 → ((𝑈𝐻𝐵) = 𝐵 → (𝐴𝑋 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍))))
3117, 22, 30sylc 62 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝐴𝑋 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍)))
32313exp 1255 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ RingOps → (𝑎𝑋 → (𝐵𝑋 → (𝐴𝑋 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍)))))
3332com15 98 . . . . . . . . . . . . . . . . . . . . 21 ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → (𝑎𝑋 → (𝐵𝑋 → (𝐴𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
3433com24 92 . . . . . . . . . . . . . . . . . . . 20 ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴𝑋 → (𝐵𝑋 → (𝑎𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
3514, 34syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝐴𝑋 → (𝐵𝑋 → (𝑎𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
3635ex 448 . . . . . . . . . . . . . . . . . 18 ((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴𝑋 → (𝐵𝑋 → (𝑎𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
3736eqcoms 2617 . . . . . . . . . . . . . . . . 17 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴𝑋 → (𝐵𝑋 → (𝑎𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
3837com25 96 . . . . . . . . . . . . . . . 16 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵) → (𝑎𝑋 → (𝐴𝑋 → (𝐵𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
3913, 38syl 17 . . . . . . . . . . . . . . 15 ((𝑎𝐻𝐴) = 𝑈 → (𝑎𝑋 → (𝐴𝑋 → (𝐵𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
4039com12 32 . . . . . . . . . . . . . 14 (𝑎𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴𝑋 → (𝐵𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
41403imp 1248 . . . . . . . . . . . . 13 ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → (𝐵𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))
4241com13 85 . . . . . . . . . . . 12 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐵𝑋 → ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))
4312, 42syl6 34 . . . . . . . . . . 11 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → (𝐵𝑋 → ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
4443com15 98 . . . . . . . . . 10 (𝑅 ∈ RingOps → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → (𝐵𝑋 → ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → 𝐵 = 𝑍)))))
45443imp1 1271 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → 𝐵 = 𝑍))
4610, 45mpd 15 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝐵 = 𝑍)
47463exp1 1274 . . . . . . 7 (𝑅 ∈ RingOps → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → (𝐵𝑋 → ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → 𝐵 = 𝑍))))
481, 47syl5com 31 . . . . . 6 ((𝐴𝐻𝐵) = 𝑍 → (𝑅 ∈ RingOps → (𝐵𝑋 → ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → 𝐵 = 𝑍))))
4948com14 93 . . . . 5 ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → (𝑅 ∈ RingOps → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍))))
50493exp 1255 . . . 4 (𝑎𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴𝑋 → (𝑅 ∈ RingOps → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍))))))
5150rexlimiv 3008 . . 3 (∃𝑎𝑋 (𝑎𝐻𝐴) = 𝑈 → (𝐴𝑋 → (𝑅 ∈ RingOps → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍)))))
5251com13 85 . 2 (𝑅 ∈ RingOps → (𝐴𝑋 → (∃𝑎𝑋 (𝑎𝐻𝐴) = 𝑈 → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍)))))
53523imp 1248 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ ∃𝑎𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wrex 2896  ran crn 5029  cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  GIdcgi 26494  RingOpscrngo 32659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fo 5796  df-fv 5798  df-riota 6489  df-ov 6530  df-1st 7036  df-2nd 7037  df-grpo 26497  df-gid 26498  df-ablo 26552  df-ass 32608  df-exid 32610  df-mgmOLD 32614  df-sgrOLD 32626  df-mndo 32632  df-rngo 32660
This theorem is referenced by:  isdrngo2  32723
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