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Theorem zerdivemp1x 34704
Description: In a unitary ring a left invertible element is not a zero divisor. See also ringinvnzdiv 19078. (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 6982 . . . . . . 7 ((𝐴𝐻𝐵) = 𝑍 → (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍))
2 simpl1 1172 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝑅 ∈ RingOps)
3 simpr1 1175 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝑎𝑋)
4 simpr3 1177 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝐴𝑋)
5 simpl3 1174 . . . . . . . . . 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 34663 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ (𝑎𝑋𝐴𝑋𝐵𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
102, 3, 4, 5, 9syl13anc 1353 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
11 eqtr 2792 . . . . . . . . . . . . 13 ((((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍))
1211ex 405 . . . . . . . . . . . 12 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)))
13 eqtr 2792 . . . . . . . . . . . . . . . . . . 19 (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
14 zerdivempx.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑍 = (GId‘𝐺)
1514, 8, 6, 7rngorz 34680 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝑎𝑋) → (𝑎𝐻𝑍) = 𝑍)
16153adant3 1113 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝑎𝐻𝑍) = 𝑍)
176rneqi 5647 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ran 𝐺 = ran (1st𝑅)
188, 17eqtri 2795 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑋 = ran (1st𝑅)
19 zerdivempx.5 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑈 = (GId‘𝐻)
207, 18, 19rngolidm 34694 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝐵𝑋) → (𝑈𝐻𝐵) = 𝐵)
21203adant2 1112 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝑈𝐻𝐵) = 𝐵)
22 simp1 1117 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
23 simp2 1118 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = 𝐵)
24 simp3 1119 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑎𝐻𝑍) = 𝑍)
2522, 23, 243eqtr3d 2815 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → 𝐵 = 𝑍)
2625a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝐴𝑋𝐵 = 𝑍))
27263exp 1100 . . . . . . . . . . . . . . . . . . . . . . . . 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 1100 . . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . 17 ((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴𝑋 → (𝐵𝑋 → (𝑎𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
3635eqcoms 2779 . . . . . . . . . . . . . . . 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 6981 . . . . . . . . . . . . . . 15 ((𝑎𝐻𝐴) = 𝑈 → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵))
3937, 38syl11 33 . . . . . . . . . . . . . 14 (𝑎𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴𝑋 → (𝐵𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
40393imp 1092 . . . . . . . . . . . . 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 1328 . . . . . . . . 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 1333 . . . . . . 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 1100 . . . 4 (𝑎𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴𝑋 → (𝑅 ∈ RingOps → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍))))))
5049rexlimiv 3218 . . 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 1092 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ ∃𝑎𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1069   = wceq 1508  wcel 2051  wrex 3082  ran crn 5404  cfv 6185  (class class class)co 6974  1st c1st 7497  2nd c2nd 7498  GIdcgi 28059  RingOpscrngo 34651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-fo 6191  df-fv 6193  df-riota 6935  df-ov 6977  df-1st 7499  df-2nd 7500  df-grpo 28062  df-gid 28063  df-ablo 28114  df-ass 34600  df-exid 34602  df-mgmOLD 34606  df-sgrOLD 34618  df-mndo 34624  df-rngo 34652
This theorem is referenced by:  isdrngo2  34715
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