Theorem ghomidOLD 38348

Description: Obsolete version of ghmid 19252 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 2761	. . . . . . 7 ⊢ ran 𝐺 = ran 𝐺
2		ghomidOLD.1	. . . . . . 7 ⊢ 𝑈 = (GId‘𝐺)
3	1, 2	grpoidcl 30673	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ ran 𝐺)
4	3	3ad2ant1 1145	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑈 ∈ ran 𝐺)
5	4, 4	jca 519	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝑈 ∈ ran 𝐺 ∧ 𝑈 ∈ ran 𝐺))
6	1	ghomlinOLD 38347	. . . 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 697	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘(𝑈𝐺𝑈)))
8	1, 2	grpolid 30675	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑈 ∈ ran 𝐺) → (𝑈𝐺𝑈) = 𝑈)
9	3, 8	mpdan 697	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝑈𝐺𝑈) = 𝑈)
10	9	fveq2d 6865	. . . 4 ⊢ (𝐺 ∈ GrpOp → (𝐹‘(𝑈𝐺𝑈)) = (𝐹‘𝑈))
11	10	3ad2ant1 1145	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘(𝑈𝐺𝑈)) = (𝐹‘𝑈))
12	7, 11	eqtrd 2796	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈))
13		eqid 2761	. . . . . . 7 ⊢ ran 𝐻 = ran 𝐻
14	1, 13	elghomOLD 38346	. . . . . 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 1489	. . . . 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 498	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:ran 𝐺⟶ran 𝐻)
17	16, 4	ffvelcdmd 7060	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) ∈ ran 𝐻)
18		ghomidOLD.2	. . . . . 6 ⊢ 𝑇 = (GId‘𝐻)
19	13, 18	grpoid 30679	. . . . 5 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝑈) ∈ ran 𝐻) → ((𝐹‘𝑈) = 𝑇 ↔ ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈)))
20	19	ex 416	. . . 4 ⊢ (𝐻 ∈ GrpOp → ((𝐹‘𝑈) ∈ ran 𝐻 → ((𝐹‘𝑈) = 𝑇 ↔ ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈))))
21	20	3ad2ant2 1146	. . 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 259	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) = 𝑇)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ran crn 5644  ⟶wf 6511  ‘cfv 6515  (class class class)co 7390  GrpOpcgr 30648  GIdcgi 30649   GrpOpHom cghomOLD 38342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-grpo 30652  df-gid 30653  df-ghomOLD 38343

This theorem is referenced by:  grpokerinj  38352  rngohom0  38431