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Mirrors > Home > MPE Home > Th. List > ghmlin | Structured version Visualization version GIF version |
Description: A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
ghmlin.x | ⊢ 𝑋 = (Base‘𝑆) |
ghmlin.a | ⊢ + = (+g‘𝑆) |
ghmlin.b | ⊢ ⨣ = (+g‘𝑇) |
Ref | Expression |
---|---|
ghmlin | ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmlin.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝑆) | |
2 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | ghmlin.a | . . . . . 6 ⊢ + = (+g‘𝑆) | |
4 | ghmlin.b | . . . . . 6 ⊢ ⨣ = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | isghm 19246 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶(Base‘𝑇) ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏))))) |
6 | 5 | simprbi 496 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋⟶(Base‘𝑇) ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)))) |
7 | 6 | simprd 495 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏))) |
8 | fvoveq1 7454 | . . . . 5 ⊢ (𝑎 = 𝑈 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑈 + 𝑏))) | |
9 | fveq2 6907 | . . . . . 6 ⊢ (𝑎 = 𝑈 → (𝐹‘𝑎) = (𝐹‘𝑈)) | |
10 | 9 | oveq1d 7446 | . . . . 5 ⊢ (𝑎 = 𝑈 → ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏))) |
11 | 8, 10 | eqeq12d 2751 | . . . 4 ⊢ (𝑎 = 𝑈 → ((𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) ↔ (𝐹‘(𝑈 + 𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)))) |
12 | oveq2 7439 | . . . . . 6 ⊢ (𝑏 = 𝑉 → (𝑈 + 𝑏) = (𝑈 + 𝑉)) | |
13 | 12 | fveq2d 6911 | . . . . 5 ⊢ (𝑏 = 𝑉 → (𝐹‘(𝑈 + 𝑏)) = (𝐹‘(𝑈 + 𝑉))) |
14 | fveq2 6907 | . . . . . 6 ⊢ (𝑏 = 𝑉 → (𝐹‘𝑏) = (𝐹‘𝑉)) | |
15 | 14 | oveq2d 7447 | . . . . 5 ⊢ (𝑏 = 𝑉 → ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))) |
16 | 13, 15 | eqeq12d 2751 | . . . 4 ⊢ (𝑏 = 𝑉 → ((𝐹‘(𝑈 + 𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)) ↔ (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉)))) |
17 | 11, 16 | rspc2v 3633 | . . 3 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉)))) |
18 | 7, 17 | mpan9 506 | . 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋)) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))) |
19 | 18 | 3impb 1114 | 1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Grpcgrp 18964 GrpHom cghm 19243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-ghm 19244 |
This theorem is referenced by: ghmid 19253 ghminv 19254 ghmsub 19255 ghmmhm 19257 ghmrn 19260 resghm 19263 ghmpreima 19269 ghmnsgima 19271 ghmnsgpreima 19272 ghmf1o 19279 ghmqusnsglem1 19311 ghmqusnsg 19313 ghmquskerlem1 19314 ghmquskerlem3 19317 lactghmga 19438 invghm 19866 ghmplusg 19879 rhmopp 20526 srngadd 20869 islmhm2 21055 rhmpreimaidl 21305 cygznlem3 21606 psgnco 21619 evpmodpmf1o 21632 ipdir 21675 evlslem1 22124 mpfind 22149 evl1addd 22361 mdetralt 22630 cpmatacl 22738 mat2pmatghm 22752 ghmcnp 24139 ply1rem 26220 dchrptlem2 27324 abliso 33024 rhmimaidl 33440 r1pquslmic 33611 dimkerim 33655 zrhcntr 33942 qqhghm 33951 qqhrhm 33952 fldhmf1 42072 aks6d1c1p3 42092 aks6d1c5lem1 42118 aks6d1c5lem2 42120 aks5lem3a 42171 evlsaddval 42555 evladdval 42562 gicabl 43088 |
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