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Theorem zerdivemp1x 37119
Description: In a unital ring a left invertible element is not a zero divisor. See also ringinvnzdiv 20190. (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 7420 . . . . . . 7 ((𝐴𝐻𝐵) = 𝑍 → (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍))
2 simpl1 1190 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝑅 ∈ RingOps)
3 simpr1 1193 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝑎𝑋)
4 simpr3 1195 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝐴𝑋)
5 simpl3 1192 . . . . . . . . . 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 37078 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ (𝑎𝑋𝐴𝑋𝐵𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
102, 3, 4, 5, 9syl13anc 1371 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
11 eqtr 2754 . . . . . . . . . . . . 13 ((((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍))
1211ex 412 . . . . . . . . . . . 12 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)))
13 eqtr 2754 . . . . . . . . . . . . . . . . . . 19 (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
14 zerdivempx.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑍 = (GId‘𝐺)
1514, 8, 6, 7rngorz 37095 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝑎𝑋) → (𝑎𝐻𝑍) = 𝑍)
16153adant3 1131 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝑎𝐻𝑍) = 𝑍)
176rneqi 5937 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ran 𝐺 = ran (1st𝑅)
188, 17eqtri 2759 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑋 = ran (1st𝑅)
19 zerdivempx.5 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑈 = (GId‘𝐻)
207, 18, 19rngolidm 37109 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝐵𝑋) → (𝑈𝐻𝐵) = 𝐵)
21203adant2 1130 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝑈𝐻𝐵) = 𝐵)
22 simp1 1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
23 simp2 1136 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = 𝐵)
24 simp3 1137 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑎𝐻𝑍) = 𝑍)
2522, 23, 243eqtr3d 2779 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → 𝐵 = 𝑍)
2625a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝐴𝑋𝐵 = 𝑍))
27263exp 1118 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → ((𝑈𝐻𝐵) = 𝐵 → ((𝑎𝐻𝑍) = 𝑍 → (𝐴𝑋𝐵 = 𝑍))))
2827com14 96 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑋 → ((𝑈𝐻𝐵) = 𝐵 → ((𝑎𝐻𝑍) = 𝑍 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍))))
2928com13 88 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎𝐻𝑍) = 𝑍 → ((𝑈𝐻𝐵) = 𝐵 → (𝐴𝑋 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍))))
3016, 21, 29sylc 65 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝐴𝑋 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍)))
31303exp 1118 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ RingOps → (𝑎𝑋 → (𝐵𝑋 → (𝐴𝑋 → ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → 𝐵 = 𝑍)))))
3231com15 101 . . . . . . . . . . . . . . . . . . . 20 ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → (𝑎𝑋 → (𝐵𝑋 → (𝐴𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
3332com24 95 . . . . . . . . . . . . . . . . . . 19 ((𝑈𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴𝑋 → (𝐵𝑋 → (𝑎𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
3413, 33syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝐴𝑋 → (𝐵𝑋 → (𝑎𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
3534ex 412 . . . . . . . . . . . . . . . . 17 ((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴𝑋 → (𝐵𝑋 → (𝑎𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
3635eqcoms 2739 . . . . . . . . . . . . . . . 16 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴𝑋 → (𝐵𝑋 → (𝑎𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
3736com25 99 . . . . . . . . . . . . . . 15 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵) → (𝑎𝑋 → (𝐴𝑋 → (𝐵𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
38 oveq1 7419 . . . . . . . . . . . . . . 15 ((𝑎𝐻𝐴) = 𝑈 → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵))
3937, 38syl11 33 . . . . . . . . . . . . . 14 (𝑎𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴𝑋 → (𝐵𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
40393imp 1110 . . . . . . . . . . . . 13 ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → (𝐵𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))
4140com13 88 . . . . . . . . . . . 12 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐵𝑋 → ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))
4212, 41syl6 35 . . . . . . . . . . 11 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → (𝐵𝑋 → ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → (𝑅 ∈ RingOps → 𝐵 = 𝑍)))))
4342com15 101 . . . . . . . . . 10 (𝑅 ∈ RingOps → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → (𝐵𝑋 → ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → 𝐵 = 𝑍)))))
44433imp1 1346 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → 𝐵 = 𝑍))
4510, 44mpd 15 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝐵 = 𝑍)
46453exp1 1351 . . . . . . 7 (𝑅 ∈ RingOps → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → (𝐵𝑋 → ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → 𝐵 = 𝑍))))
471, 46syl5com 31 . . . . . 6 ((𝐴𝐻𝐵) = 𝑍 → (𝑅 ∈ RingOps → (𝐵𝑋 → ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → 𝐵 = 𝑍))))
4847com14 96 . . . . 5 ((𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋) → (𝑅 ∈ RingOps → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍))))
49483exp 1118 . . . 4 (𝑎𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴𝑋 → (𝑅 ∈ RingOps → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍))))))
5049rexlimiv 3147 . . 3 (∃𝑎𝑋 (𝑎𝐻𝐴) = 𝑈 → (𝐴𝑋 → (𝑅 ∈ RingOps → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍)))))
5150com13 88 . 2 (𝑅 ∈ RingOps → (𝐴𝑋 → (∃𝑎𝑋 (𝑎𝐻𝐴) = 𝑈 → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍)))))
52513imp 1110 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ ∃𝑎𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  wrex 3069  ran crn 5678  cfv 6544  (class class class)co 7412  1st c1st 7976  2nd c2nd 7977  GIdcgi 30007  RingOpscrngo 37066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-riota 7368  df-ov 7415  df-1st 7978  df-2nd 7979  df-grpo 30010  df-gid 30011  df-ablo 30062  df-ass 37015  df-exid 37017  df-mgmOLD 37021  df-sgrOLD 37033  df-mndo 37039  df-rngo 37067
This theorem is referenced by:  isdrngo2  37130
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