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Theorem cmpidelt 36368
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 36366 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → 𝑈 = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
43eqcomd 2739 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈)
51, 2iorlid 36367 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → 𝑈𝑋)
61exidu1 36365 . . . 4 (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
7 oveq1 7368 . . . . . . 7 (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
87eqeq1d 2735 . . . . . 6 (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
98ovanraleqv 7385 . . . . 5 (𝑢 = 𝑈 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
109riota2 7343 . . . 4 ((𝑈𝑋 ∧ ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → (∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈))
115, 6, 10syl2anc 585 . . 3 (𝐺 ∈ (Magma ∩ ExId ) → (∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) = 𝑈))
124, 11mpbird 257 . 2 (𝐺 ∈ (Magma ∩ ExId ) → ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥))
13 oveq2 7369 . . . . 5 (𝑥 = 𝐴 → (𝑈𝐺𝑥) = (𝑈𝐺𝐴))
14 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
1513, 14eqeq12d 2749 . . . 4 (𝑥 = 𝐴 → ((𝑈𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝐴) = 𝐴))
16 oveq1 7368 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑈) = (𝐴𝐺𝑈))
1716, 14eqeq12d 2749 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐺𝑈) = 𝑥 ↔ (𝐴𝐺𝑈) = 𝐴))
1815, 17anbi12d 632 . . 3 (𝑥 = 𝐴 → (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)))
1918rspccva 3582 . 2 ((∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ 𝐴𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))
2012, 19sylan 581 1 ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  ∃!wreu 3350  cin 3913  ran crn 5638  cfv 6500  crio 7316  (class class class)co 7361  GIdcgi 29481   ExId cexid 36353  Magmacmagm 36357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-riota 7317  df-ov 7364  df-gid 29485  df-exid 36354  df-mgmOLD 36358
This theorem is referenced by:  exidreslem  36386  rngoidmlem  36445
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