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Mirrors > Home > MPE Home > Th. List > Mathboxes > ghomidOLD | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ghomidOLD.1 | ⊢ 𝑈 = (GId‘𝐺) |
ghomidOLD.2 | ⊢ 𝑇 = (GId‘𝐻) |
Ref | Expression |
---|---|
ghomidOLD | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) = 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . . . . 7 ⊢ ran 𝐺 = ran 𝐺 | |
2 | ghomidOLD.1 | . . . . . . 7 ⊢ 𝑈 = (GId‘𝐺) | |
3 | 1, 2 | grpoidcl 26752 | . . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ ran 𝐺) |
4 | 3 | 3ad2ant1 1075 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑈 ∈ ran 𝐺) |
5 | 4, 4 | jca 553 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝑈 ∈ ran 𝐺 ∧ 𝑈 ∈ ran 𝐺)) |
6 | 1 | ghomlinOLD 32857 | . . . 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 699 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘(𝑈𝐺𝑈))) |
8 | 1, 2 | grpolid 26754 | . . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑈 ∈ ran 𝐺) → (𝑈𝐺𝑈) = 𝑈) |
9 | 3, 8 | mpdan 699 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝑈𝐺𝑈) = 𝑈) |
10 | 9 | fveq2d 6107 | . . . 4 ⊢ (𝐺 ∈ GrpOp → (𝐹‘(𝑈𝐺𝑈)) = (𝐹‘𝑈)) |
11 | 10 | 3ad2ant1 1075 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘(𝑈𝐺𝑈)) = (𝐹‘𝑈)) |
12 | 7, 11 | eqtrd 2644 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈)) |
13 | eqid 2610 | . . . . . . 7 ⊢ ran 𝐻 = ran 𝐻 | |
14 | 1, 13 | elghomOLD 32856 | . . . . . 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 1424 | . . . . 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 474 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:ran 𝐺⟶ran 𝐻) |
17 | 16, 4 | ffvelrnd 6268 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) ∈ ran 𝐻) |
18 | ghomidOLD.2 | . . . . . 6 ⊢ 𝑇 = (GId‘𝐻) | |
19 | 13, 18 | grpoid 26758 | . . . . 5 ⊢ ((𝐻 ∈ GrpOp ∧ (𝐹‘𝑈) ∈ ran 𝐻) → ((𝐹‘𝑈) = 𝑇 ↔ ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈))) |
20 | 19 | ex 449 | . . . 4 ⊢ (𝐻 ∈ GrpOp → ((𝐹‘𝑈) ∈ ran 𝐻 → ((𝐹‘𝑈) = 𝑇 ↔ ((𝐹‘𝑈)𝐻(𝐹‘𝑈)) = (𝐹‘𝑈)))) |
21 | 20 | 3ad2ant2 1076 | . . 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 246 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹‘𝑈) = 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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