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Theorem zerdivemp1x 37934
Description: In a unital ring a left invertible element is not a zero divisor. See also ringinvnzdiv 20315. (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 7439 . . . . . . 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 37893 . . . . . . . . . 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 2758 . . . . . . . . . . . . 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 2758 . . . . . . . . . . . . . . . . . . 19 (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
14 zerdivempx.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑍 = (GId‘𝐺)
1514, 8, 6, 7rngorz 37910 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝑎𝑋) → (𝑎𝐻𝑍) = 𝑍)
16153adant3 1131 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝑎𝐻𝑍) = 𝑍)
176rneqi 5951 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ran 𝐺 = ran (1st𝑅)
188, 17eqtri 2763 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑋 = ran (1st𝑅)
19 zerdivempx.5 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑈 = (GId‘𝐻)
207, 18, 19rngolidm 37924 . . . . . . . . . . . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . 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 3146 . . 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 1537  wcel 2106  wrex 3068  ran crn 5690  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  GIdcgi 30519  RingOpscrngo 37881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-riota 7388  df-ov 7434  df-1st 8013  df-2nd 8014  df-grpo 30522  df-gid 30523  df-ablo 30574  df-ass 37830  df-exid 37832  df-mgmOLD 37836  df-sgrOLD 37848  df-mndo 37854  df-rngo 37882
This theorem is referenced by:  isdrngo2  37945
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