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Theorem zerdivemp1x 35106
Description: In a unitary ring a left invertible element is not a zero divisor. See also ringinvnzdiv 19272. (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 7153 . . . . . . 7 ((𝐴𝐻𝐵) = 𝑍 → (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍))
2 simpl1 1183 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝑅 ∈ RingOps)
3 simpr1 1186 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝑎𝑋)
4 simpr3 1188 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → 𝐴𝑋)
5 simpl3 1185 . . . . . . . . . 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 35065 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ (𝑎𝑋𝐴𝑋𝐵𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
102, 3, 4, 5, 9syl13anc 1364 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) ∧ 𝐵𝑋) ∧ (𝑎𝑋 ∧ (𝑎𝐻𝐴) = 𝑈𝐴𝑋)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)))
11 eqtr 2838 . . . . . . . . . . . . 13 ((((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) ∧ (𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍)) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍))
1211ex 413 . . . . . . . . . . . 12 (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻(𝐴𝐻𝐵)) → ((𝑎𝐻(𝐴𝐻𝐵)) = (𝑎𝐻𝑍) → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)))
13 eqtr 2838 . . . . . . . . . . . . . . . . . . 19 (((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) ∧ ((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍)) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
14 zerdivempx.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑍 = (GId‘𝐺)
1514, 8, 6, 7rngorz 35082 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝑎𝑋) → (𝑎𝐻𝑍) = 𝑍)
16153adant3 1124 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝑎𝐻𝑍) = 𝑍)
176rneqi 5800 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ran 𝐺 = ran (1st𝑅)
188, 17eqtri 2841 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑋 = ran (1st𝑅)
19 zerdivempx.5 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑈 = (GId‘𝐻)
207, 18, 19rngolidm 35096 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑅 ∈ RingOps ∧ 𝐵𝑋) → (𝑈𝐻𝐵) = 𝐵)
21203adant2 1123 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ 𝑎𝑋𝐵𝑋) → (𝑈𝐻𝐵) = 𝐵)
22 simp1 1128 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = (𝑎𝐻𝑍))
23 simp2 1129 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑈𝐻𝐵) = 𝐵)
24 simp3 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝑎𝐻𝑍) = 𝑍)
2522, 23, 243eqtr3d 2861 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → 𝐵 = 𝑍)
2625a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑈𝐻𝐵) = (𝑎𝐻𝑍) ∧ (𝑈𝐻𝐵) = 𝐵 ∧ (𝑎𝐻𝑍) = 𝑍) → (𝐴𝑋𝐵 = 𝑍))
27263exp 1111 . . . . . . . . . . . . . . . . . . . . . . . . 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 1111 . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . 17 ((𝑈𝐻𝐵) = ((𝑎𝐻𝐴)𝐻𝐵) → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝐴𝑋 → (𝐵𝑋 → (𝑎𝑋 → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
3635eqcoms 2826 . . . . . . . . . . . . . . . 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 7152 . . . . . . . . . . . . . . 15 ((𝑎𝐻𝐴) = 𝑈 → ((𝑎𝐻𝐴)𝐻𝐵) = (𝑈𝐻𝐵))
3937, 38syl11 33 . . . . . . . . . . . . . 14 (𝑎𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴𝑋 → (𝐵𝑋 → (((𝑎𝐻𝐴)𝐻𝐵) = (𝑎𝐻𝑍) → (𝑅 ∈ RingOps → 𝐵 = 𝑍))))))
40393imp 1103 . . . . . . . . . . . . 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 1339 . . . . . . . . 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 1344 . . . . . . 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 1111 . . . 4 (𝑎𝑋 → ((𝑎𝐻𝐴) = 𝑈 → (𝐴𝑋 → (𝑅 ∈ RingOps → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍))))))
5049rexlimiv 3277 . . 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 1103 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ ∃𝑎𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  ran crn 5549  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  GIdcgi 28194  RingOpscrngo 35053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-riota 7103  df-ov 7148  df-1st 7678  df-2nd 7679  df-grpo 28197  df-gid 28198  df-ablo 28249  df-ass 35002  df-exid 35004  df-mgmOLD 35008  df-sgrOLD 35020  df-mndo 35026  df-rngo 35054
This theorem is referenced by:  isdrngo2  35117
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