Theorem ghomidOLD 37830

Description: Obsolete version of ghmid 19208 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 2734	. . . . . . 7 ⊢ ran 𝐺 = ran 𝐺
2		ghomidOLD.1	. . . . . . 7 ⊢ 𝑈 = (GId‘𝐺)
3	1, 2	grpoidcl 30460	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ ran 𝐺)
4	3	3ad2ant1 1133	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑈 ∈ ran 𝐺)
5	4, 4	jca 511	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝑈 ∈ ran 𝐺 ∧ 𝑈 ∈ ran 𝐺))
6	1	ghomlinOLD 37829	. . . 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 687	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
8	1, 2	grpolid 30462	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑈 ∈ ran 𝐺) → (𝑈𝐺𝑈) = 𝑈)
9	3, 8	mpdan 687	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝑈𝐺𝑈) = 𝑈)
10	9	fveq2d 6889	. . . 4 ⊢ (𝐺 ∈ GrpOp → (𝐹‘(𝑈𝐺𝑈)) = (𝐹‘𝑈))
11	10	3ad2ant1 1133	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘(𝑈𝐺𝑈)) = (𝐹‘𝑈))
12	7, 11	eqtrd 2769	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈))
13		eqid 2734	. . . . . . 7 ⊢ ran 𝐻 = ran 𝐻
14	1, 13	elghomOLD 37828	. . . . . 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 1470	. . . . 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 494	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:ran 𝐺⟶ran 𝐻)
17	16, 4	ffvelcdmd 7084	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) ∈ ran 𝐻)
18		ghomidOLD.2	. . . . . 6 ⊢ 𝑇 = (GId‘𝐻)
19	13, 18	grpoid 30466	. . . . 5 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝑈) ∈ ran 𝐻) → ((𝐹‘𝑈) = 𝑇 ↔ ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈)))
20	19	ex 412	. . . 4 ⊢ (𝐻 ∈ GrpOp → ((𝐹‘𝑈) ∈ ran 𝐻 → ((𝐹‘𝑈) = 𝑇 ↔ ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈))))
21	20	3ad2ant2 1134	. . 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 257	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) = 𝑇)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ran crn 5666  ⟶wf 6536  ‘cfv 6540  (class class class)co 7412  GrpOpcgr 30435  GIdcgi 30436   GrpOpHom cghomOLD 37824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-grpo 30439  df-gid 30440  df-ghomOLD 37825

This theorem is referenced by:  grpokerinj  37834  rngohom0  37913