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Theorem rngoid 33333
Description: The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1 𝐺 = (1st𝑅)
ringi.2 𝐻 = (2nd𝑅)
ringi.3 𝑋 = ran 𝐺
Assertion
Ref Expression
rngoid ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑢𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))
Distinct variable groups:   𝑢,𝐺   𝑢,𝐻   𝑢,𝑋   𝑢,𝐴   𝑢,𝑅

Proof of Theorem rngoid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . 5 𝐺 = (1st𝑅)
2 ringi.2 . . . . 5 𝐻 = (2nd𝑅)
3 ringi.3 . . . . 5 𝑋 = ran 𝐺
41, 2, 3rngoi 33330 . . . 4 (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑢𝑋𝑥𝑋𝑦𝑋 (((𝑢𝐻𝑥)𝐻𝑦) = (𝑢𝐻(𝑥𝐻𝑦)) ∧ (𝑢𝐻(𝑥𝐺𝑦)) = ((𝑢𝐻𝑥)𝐺(𝑢𝐻𝑦)) ∧ ((𝑢𝐺𝑥)𝐻𝑦) = ((𝑢𝐻𝑦)𝐺(𝑥𝐻𝑦))) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))))
54simprrd 796 . . 3 (𝑅 ∈ RingOps → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
6 r19.12 3056 . . 3 (∃𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → ∀𝑥𝑋𝑢𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
75, 6syl 17 . 2 (𝑅 ∈ RingOps → ∀𝑥𝑋𝑢𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
8 oveq2 6612 . . . . . 6 (𝑥 = 𝐴 → (𝑢𝐻𝑥) = (𝑢𝐻𝐴))
9 id 22 . . . . . 6 (𝑥 = 𝐴𝑥 = 𝐴)
108, 9eqeq12d 2636 . . . . 5 (𝑥 = 𝐴 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑢𝐻𝐴) = 𝐴))
11 oveq1 6611 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐻𝑢) = (𝐴𝐻𝑢))
1211, 9eqeq12d 2636 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐻𝑢) = 𝑥 ↔ (𝐴𝐻𝑢) = 𝐴))
1310, 12anbi12d 746 . . . 4 (𝑥 = 𝐴 → (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴)))
1413rexbidv 3045 . . 3 (𝑥 = 𝐴 → (∃𝑢𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∃𝑢𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴)))
1514rspccva 3294 . 2 ((∀𝑥𝑋𝑢𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ 𝐴𝑋) → ∃𝑢𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))
167, 15sylan 488 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ∃𝑢𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908   × cxp 5072  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  AbelOpcablo 27247  RingOpscrngo 33325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-1st 7113  df-2nd 7114  df-rngo 33326
This theorem is referenced by:  rngo2  33338
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