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)) → (𝐹‘𝑈) = 𝑇)

Proof of Theorem ghomidOLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2726	. . . . . . 7 ⊢ ran 𝐺 = ran 𝐺
2		ghomidOLD.1	. . . . . . 7 ⊢ 𝑈 = (GId‘𝐺)
3	1, 2	grpoidcl 30450	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ ran 𝐺)
4	3	3ad2ant1 1130	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑈 ∈ ran 𝐺)
5	4, 4	jca 510	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝑈 ∈ ran 𝐺 ∧ 𝑈 ∈ ran 𝐺))
6	1	ghomlinOLD 37591	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑈 ∈ ran 𝐺 ∧ 𝑈 ∈ ran 𝐺)) → ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
7	5, 6	mpdan 685	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
8	1, 2	grpolid 30452	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑈 ∈ ran 𝐺) → (𝑈𝐺𝑈) = 𝑈)
9	3, 8	mpdan 685	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝑈𝐺𝑈) = 𝑈)
10	9	fveq2d 6907	. . . 4 ⊢ (𝐺 ∈ GrpOp → (𝐹‘(𝑈𝐺𝑈)) = (𝐹‘𝑈))
11	10	3ad2ant1 1130	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘(𝑈𝐺𝑈)) = (𝐹‘𝑈))
12	7, 11	eqtrd 2766	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈))
13		eqid 2726	. . . . . . 7 ⊢ ran 𝐻 = ran 𝐻
14	1, 13	elghomOLD 37590	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
15	14	biimp3a 1466	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
16	15	simpld 493	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:ran 𝐺⟶ran 𝐻)
17	16, 4	ffvelcdmd 7101	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) ∈ ran 𝐻)
18		ghomidOLD.2	. . . . . 6 ⊢ 𝑇 = (GId‘𝐻)
19	13, 18	grpoid 30456	. . . . 5 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝑈) ∈ ran 𝐻) → ((𝐹‘𝑈) = 𝑇 ↔ ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈)))
20	19	ex 411	. . . 4 ⊢ (𝐻 ∈ GrpOp → ((𝐹‘𝑈) ∈ ran 𝐻 → ((𝐹‘𝑈) = 𝑇 ↔ ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈))))
21	20	3ad2ant2 1131	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈) ∈ ran 𝐻 → ((𝐹‘𝑈) = 𝑇 ↔ ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈))))
22	17, 21	mpd 15	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈) = 𝑇 ↔ ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈)))
23	12, 22	mpbird 256	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) = 𝑇)

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