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Theorem ghomidOLD 34230
Description: Obsolete version of ghmid 18017 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 2825 . . . . . . 7 ran 𝐺 = ran 𝐺
2 ghomidOLD.1 . . . . . . 7 𝑈 = (GId‘𝐺)
31, 2grpoidcl 27924 . . . . . 6 (𝐺 ∈ GrpOp → 𝑈 ∈ ran 𝐺)
433ad2ant1 1169 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑈 ∈ ran 𝐺)
54, 4jca 509 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝑈 ∈ ran 𝐺𝑈 ∈ ran 𝐺))
61ghomlinOLD 34229 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑈 ∈ ran 𝐺𝑈 ∈ ran 𝐺)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
75, 6mpdan 680 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
81, 2grpolid 27926 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑈 ∈ ran 𝐺) → (𝑈𝐺𝑈) = 𝑈)
93, 8mpdan 680 . . . . 5 (𝐺 ∈ GrpOp → (𝑈𝐺𝑈) = 𝑈)
109fveq2d 6437 . . . 4 (𝐺 ∈ GrpOp → (𝐹‘(𝑈𝐺𝑈)) = (𝐹𝑈))
11103ad2ant1 1169 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘(𝑈𝐺𝑈)) = (𝐹𝑈))
127, 11eqtrd 2861 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈))
13 eqid 2825 . . . . . . 7 ran 𝐻 = ran 𝐻
141, 13elghomOLD 34228 . . . . . 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 1599 . . . . 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 490 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:ran 𝐺⟶ran 𝐻)
1716, 4ffvelrnd 6609 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑈) ∈ ran 𝐻)
18 ghomidOLD.2 . . . . . 6 𝑇 = (GId‘𝐻)
1913, 18grpoid 27930 . . . . 5 ((𝐻 ∈ GrpOp ∧ (𝐹𝑈) ∈ ran 𝐻) → ((𝐹𝑈) = 𝑇 ↔ ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈)))
2019ex 403 . . . 4 (𝐻 ∈ GrpOp → ((𝐹𝑈) ∈ ran 𝐻 → ((𝐹𝑈) = 𝑇 ↔ ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈))))
21203ad2ant2 1170 . . 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 249 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑈) = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3117  ran crn 5343  wf 6119  cfv 6123  (class class class)co 6905  GrpOpcgr 27899  GIdcgi 27900   GrpOpHom cghomOLD 34224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-grpo 27903  df-gid 27904  df-ghomOLD 34225
This theorem is referenced by:  grpokerinj  34234  rngohom0  34313
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