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Theorem cmpidelt 33971
Description: A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmpidelt.1 𝑋 = ran 𝐺
cmpidelt.2 𝑈 = (GId‘𝐺)
Assertion
Ref Expression
cmpidelt ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))

Proof of Theorem cmpidelt
Dummy variables 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmpidelt.1 . . . . 5 𝑋 = ran 𝐺
2 cmpidelt.2 . . . . 5 𝑈 = (GId‘𝐺)
31, 2idrval 33969 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
43eqcomd 2766 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈)
51, 2iorlid 33970 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → 𝑈𝑋)
61exidu1 33968 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
7 oveq1 6820 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
87eqeq1d 2762 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
9 oveq2 6821 . . . . . . . 8 (𝑢 = 𝑈 → (𝑥𝐺𝑢) = (𝑥𝐺𝑈))
109eqeq1d 2762 . . . . . . 7 (𝑢 = 𝑈 → ((𝑥𝐺𝑢) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
118, 10anbi12d 749 . . . . . 6 (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
1211ralbidv 3124 . . . . 5 (𝑢 = 𝑈 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
1312riota2 6796 . . . 4 ((𝑈𝑋 ∧ ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → (∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈))
145, 6, 13syl2anc 696 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → (∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈))
154, 14mpbird 247 . 2 (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
16 oveq2 6821 . . . . 5 (𝑥 = 𝐴 → (𝑈𝐺𝑥) = (𝑈𝐺𝐴))
17 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
1816, 17eqeq12d 2775 . . . 4 (𝑥 = 𝐴 → ((𝑈𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝐴) = 𝐴))
19 oveq1 6820 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑈) = (𝐴𝐺𝑈))
2019, 17eqeq12d 2775 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐺𝑈) = 𝑥 ↔ (𝐴𝐺𝑈) = 𝐴))
2118, 20anbi12d 749 . . 3 (𝑥 = 𝐴 → (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)))
2221rspccva 3448 . 2 ((∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ 𝐴𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))
2315, 22sylan 489 1 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  ∃!wreu 3052  cin 3714  ran crn 5267  cfv 6049  crio 6773  (class class class)co 6813  GIdcgi 27653   ExId cexid 33956  Magmacmagm 33960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-riota 6774  df-ov 6816  df-gid 27657  df-exid 33957  df-mgmOLD 33961
This theorem is referenced by:  exidreslem  33989  rngoidmlem  34048
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