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Theorem ghomidOLD 37592
Description: Obsolete version of ghmid 19218 as of 15-Mar-2020. A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ghomidOLD.1 𝑈 = (GId‘𝐺)
ghomidOLD.2 𝑇 = (GId‘𝐻)
Assertion
Ref Expression
ghomidOLD ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑈) = 𝑇)

Proof of Theorem ghomidOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . . . . . 7 ran 𝐺 = ran 𝐺
2 ghomidOLD.1 . . . . . . 7 𝑈 = (GId‘𝐺)
31, 2grpoidcl 30450 . . . . . 6 (𝐺 ∈ GrpOp → 𝑈 ∈ ran 𝐺)
433ad2ant1 1130 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑈 ∈ ran 𝐺)
54, 4jca 510 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝑈 ∈ ran 𝐺𝑈 ∈ ran 𝐺))
61ghomlinOLD 37591 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑈 ∈ ran 𝐺𝑈 ∈ ran 𝐺)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
75, 6mpdan 685 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
81, 2grpolid 30452 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑈 ∈ ran 𝐺) → (𝑈𝐺𝑈) = 𝑈)
93, 8mpdan 685 . . . . 5 (𝐺 ∈ GrpOp → (𝑈𝐺𝑈) = 𝑈)
109fveq2d 6907 . . . 4 (𝐺 ∈ GrpOp → (𝐹‘(𝑈𝐺𝑈)) = (𝐹𝑈))
11103ad2ant1 1130 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘(𝑈𝐺𝑈)) = (𝐹𝑈))
127, 11eqtrd 2766 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈))
13 eqid 2726 . . . . . . 7 ran 𝐻 = ran 𝐻
141, 13elghomOLD 37590 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
1514biimp3a 1466 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
1615simpld 493 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:ran 𝐺⟶ran 𝐻)
1716, 4ffvelcdmd 7101 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑈) ∈ ran 𝐻)
18 ghomidOLD.2 . . . . . 6 𝑇 = (GId‘𝐻)
1913, 18grpoid 30456 . . . . 5 ((𝐻 ∈ GrpOp ∧ (𝐹𝑈) ∈ ran 𝐻) → ((𝐹𝑈) = 𝑇 ↔ ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈)))
2019ex 411 . . . 4 (𝐻 ∈ GrpOp → ((𝐹𝑈) ∈ ran 𝐻 → ((𝐹𝑈) = 𝑇 ↔ ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈))))
21203ad2ant2 1131 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹𝑈) ∈ ran 𝐻 → ((𝐹𝑈) = 𝑇 ↔ ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈))))
2217, 21mpd 15 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹𝑈) = 𝑇 ↔ ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈)))
2312, 22mpbird 256 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑈) = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  ran crn 5685  wf 6552  cfv 6556  (class class class)co 7426  GrpOpcgr 30425  GIdcgi 30426   GrpOpHom cghomOLD 37586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-grpo 30429  df-gid 30430  df-ghomOLD 37587
This theorem is referenced by:  grpokerinj  37596  rngohom0  37675
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