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Theorem ghomidOLD 32858
 Description: Obsolete version of ghmid 17489 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 2610 . . . . . . 7 ran 𝐺 = ran 𝐺
2 ghomidOLD.1 . . . . . . 7 𝑈 = (GId‘𝐺)
31, 2grpoidcl 26752 . . . . . 6 (𝐺 ∈ GrpOp → 𝑈 ∈ ran 𝐺)
433ad2ant1 1075 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑈 ∈ ran 𝐺)
54, 4jca 553 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝑈 ∈ ran 𝐺𝑈 ∈ ran 𝐺))
61ghomlinOLD 32857 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑈 ∈ ran 𝐺𝑈 ∈ ran 𝐺)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
75, 6mpdan 699 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
81, 2grpolid 26754 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑈 ∈ ran 𝐺) → (𝑈𝐺𝑈) = 𝑈)
93, 8mpdan 699 . . . . 5 (𝐺 ∈ GrpOp → (𝑈𝐺𝑈) = 𝑈)
109fveq2d 6107 . . . 4 (𝐺 ∈ GrpOp → (𝐹‘(𝑈𝐺𝑈)) = (𝐹𝑈))
11103ad2ant1 1075 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘(𝑈𝐺𝑈)) = (𝐹𝑈))
127, 11eqtrd 2644 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈))
13 eqid 2610 . . . . . . 7 ran 𝐻 = ran 𝐻
141, 13elghomOLD 32856 . . . . . 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 1424 . . . . 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 474 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:ran 𝐺⟶ran 𝐻)
1716, 4ffvelrnd 6268 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑈) ∈ ran 𝐻)
18 ghomidOLD.2 . . . . . 6 𝑇 = (GId‘𝐻)
1913, 18grpoid 26758 . . . . 5 ((𝐻 ∈ GrpOp ∧ (𝐹𝑈) ∈ ran 𝐻) → ((𝐹𝑈) = 𝑇 ↔ ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈)))
2019ex 449 . . . 4 (𝐻 ∈ GrpOp → ((𝐹𝑈) ∈ ran 𝐻 → ((𝐹𝑈) = 𝑇 ↔ ((𝐹𝑈)𝐻(𝐹𝑈)) = (𝐹𝑈))))
21203ad2ant2 1076 . . 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 246 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑈) = 𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  GrpOpcgr 26727  GIdcgi 26728   GrpOpHom cghomOLD 32852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-grpo 26731  df-gid 26732  df-ghomOLD 32853 This theorem is referenced by:  grpokerinj  32862  rngohom0  32941
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