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Mirrors > Home > MPE Home > Th. List > isghmd | Structured version Visualization version GIF version |
Description: Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.) |
Ref | Expression |
---|---|
isghmd.x | ⊢ 𝑋 = (Base‘𝑆) |
isghmd.y | ⊢ 𝑌 = (Base‘𝑇) |
isghmd.a | ⊢ + = (+g‘𝑆) |
isghmd.b | ⊢ ⨣ = (+g‘𝑇) |
isghmd.s | ⊢ (𝜑 → 𝑆 ∈ Grp) |
isghmd.t | ⊢ (𝜑 → 𝑇 ∈ Grp) |
isghmd.f | ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
isghmd.l | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
Ref | Expression |
---|---|
isghmd | ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isghmd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ Grp) | |
2 | isghmd.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ Grp) | |
3 | 1, 2 | jca 553 | . 2 ⊢ (𝜑 → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp)) |
4 | isghmd.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) | |
5 | isghmd.l | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) | |
6 | 5 | ralrimivva 2954 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
7 | 4, 6 | jca 553 | . 2 ⊢ (𝜑 → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))) |
8 | isghmd.x | . . 3 ⊢ 𝑋 = (Base‘𝑆) | |
9 | isghmd.y | . . 3 ⊢ 𝑌 = (Base‘𝑇) | |
10 | isghmd.a | . . 3 ⊢ + = (+g‘𝑆) | |
11 | isghmd.b | . . 3 ⊢ ⨣ = (+g‘𝑇) | |
12 | 8, 9, 10, 11 | isghm 17483 | . 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))) |
13 | 3, 7, 12 | sylanbrc 695 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Grpcgrp 17245 GrpHom cghm 17480 |
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-ov 6552 df-oprab 6553 df-mpt2 6554 df-ghm 17481 |
This theorem is referenced by: ghmmhmb 17494 resghm 17499 conjghm 17514 qusghm 17520 invoppggim 17613 galactghm 17646 pj1ghm 17939 frgpup1 18011 mulgghm 18057 ghmfghm 18059 invghm 18062 ghmplusg 18072 ringlghm 18427 ringrghm 18428 isrhmd 18552 lmodvsghm 18747 pwssplit2 18881 asclghm 19159 evlslem1 19336 cygznlem3 19737 psgnghm 19745 frlmup1 19956 mat1ghm 20108 scmatghm 20158 mat2pmatghm 20354 pm2mpghm 20440 reefgim 24008 qqhghm 29360 imasgim 36688 isrnghmd 41692 amgmlemALT 42358 |
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