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Theorem ghomidOLD 35048
Description: Obsolete version of ghmid 18302 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 2818 . . . . . . 7 ran 𝐺 = ran 𝐺
2 ghomidOLD.1 . . . . . . 7 𝑈 = (GId‘𝐺)
31, 2grpoidcl 28218 . . . . . 6 (𝐺 ∈ GrpOp → 𝑈 ∈ ran 𝐺)
433ad2ant1 1125 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑈 ∈ ran 𝐺)
54, 4jca 512 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝑈 ∈ ran 𝐺𝑈 ∈ ran 𝐺))
61ghomlinOLD 35047 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑈 ∈ ran 𝐺𝑈 ∈ ran 𝐺)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
75, 6mpdan 683 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
81, 2grpolid 28220 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑈 ∈ ran 𝐺) → (𝑈𝐺𝑈) = 𝑈)
93, 8mpdan 683 . . . . 5 (𝐺 ∈ GrpOp → (𝑈𝐺𝑈) = 𝑈)
109fveq2d 6667 . . . 4 (𝐺 ∈ GrpOp → (𝐹‘(𝑈𝐺𝑈)) = (𝐹𝑈))
11103ad2ant1 1125 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘(𝑈𝐺𝑈)) = (𝐹𝑈))
127, 11eqtrd 2853 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈))
13 eqid 2818 . . . . . . 7 ran 𝐻 = ran 𝐻
141, 13elghomOLD 35046 . . . . . 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 1460 . . . . 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 495 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:ran 𝐺⟶ran 𝐻)
1716, 4ffvelrnd 6844 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑈) ∈ ran 𝐻)
18 ghomidOLD.2 . . . . . 6 𝑇 = (GId‘𝐻)
1913, 18grpoid 28224 . . . . 5 ((𝐻 ∈ GrpOp ∧ (𝐹𝑈) ∈ ran 𝐻) → ((𝐹𝑈) = 𝑇 ↔ ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈)))
2019ex 413 . . . 4 (𝐻 ∈ GrpOp → ((𝐹𝑈) ∈ ran 𝐻 → ((𝐹𝑈) = 𝑇 ↔ ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈))))
21203ad2ant2 1126 . . 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 258 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑈) = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7145  GrpOpcgr 28193  GIdcgi 28194   GrpOpHom cghomOLD 35042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-grpo 28197  df-gid 28198  df-ghomOLD 35043
This theorem is referenced by:  grpokerinj  35052  rngohom0  35131
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