Theorem ghomidOLD 37890

Description: Obsolete version of ghmid 19262 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 2737	. . . . . . 7 ⊢ ran 𝐺 = ran 𝐺
2		ghomidOLD.1	. . . . . . 7 ⊢ 𝑈 = (GId‘𝐺)
3	1, 2	grpoidcl 30559	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ ran 𝐺)
4	3	3ad2ant1 1134	. . . . 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 37889	. . . 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 30561	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑈 ∈ ran 𝐺) → (𝑈𝐺𝑈) = 𝑈)
9	3, 8	mpdan 687	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝑈𝐺𝑈) = 𝑈)
10	9	fveq2d 6918	. . . 4 ⊢ (𝐺 ∈ GrpOp → (𝐹‘(𝑈𝐺𝑈)) = (𝐹‘𝑈))
11	10	3ad2ant1 1134	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘(𝑈𝐺𝑈)) = (𝐹‘𝑈))
12	7, 11	eqtrd 2777	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈))
13		eqid 2737	. . . . . . 7 ⊢ ran 𝐻 = ran 𝐻
14	1, 13	elghomOLD 37888	. . . . . 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 7112	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) ∈ ran 𝐻)
18		ghomidOLD.2	. . . . . 6 ⊢ 𝑇 = (GId‘𝐻)
19	13, 18	grpoid 30565	. . . . 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 1135	. . 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 1087   = wceq 1539   ∈ wcel 2108  ∀wral 3061  ran crn 5694  ⟶wf 6565  ‘cfv 6569  (class class class)co 7438  GrpOpcgr 30534  GIdcgi 30535   GrpOpHom cghomOLD 37884

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-riota 7395  df-ov 7441  df-oprab 7442  df-mpo 7443  df-grpo 30538  df-gid 30539  df-ghomOLD 37885

This theorem is referenced by:  grpokerinj  37894  rngohom0  37973