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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 2732 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | ghmlin.a | . . . . . 6 ⊢ + = (+g‘𝑆) | |
4 | ghmlin.b | . . . . . 6 ⊢ ⨣ = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | isghm 19130 | . . . . 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 497 | . . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋⟶(Base‘𝑇) ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)))) |
7 | 6 | simprd 496 | . . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏))) |
8 | fvoveq1 7434 | . . . . 5 ⊢ (𝑎 = 𝑈 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑈 + 𝑏))) | |
9 | fveq2 6891 | . . . . . 6 ⊢ (𝑎 = 𝑈 → (𝐹‘𝑎) = (𝐹‘𝑈)) | |
10 | 9 | oveq1d 7426 | . . . . 5 ⊢ (𝑎 = 𝑈 → ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏))) |
11 | 8, 10 | eqeq12d 2748 | . . . 4 ⊢ (𝑎 = 𝑈 → ((𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) ↔ (𝐹‘(𝑈 + 𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)))) |
12 | oveq2 7419 | . . . . . 6 ⊢ (𝑏 = 𝑉 → (𝑈 + 𝑏) = (𝑈 + 𝑉)) | |
13 | 12 | fveq2d 6895 | . . . . 5 ⊢ (𝑏 = 𝑉 → (𝐹‘(𝑈 + 𝑏)) = (𝐹‘(𝑈 + 𝑉))) |
14 | fveq2 6891 | . . . . . 6 ⊢ (𝑏 = 𝑉 → (𝐹‘𝑏) = (𝐹‘𝑉)) | |
15 | 14 | oveq2d 7427 | . . . . 5 ⊢ (𝑏 = 𝑉 → ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))) |
16 | 13, 15 | eqeq12d 2748 | . . . 4 ⊢ (𝑏 = 𝑉 → ((𝐹‘(𝑈 + 𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)) ↔ (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉)))) |
17 | 11, 16 | rspc2v 3622 | . . 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 507 | . 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋)) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))) |
19 | 18 | 3impb 1115 | 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 Grpcgrp 18855 GrpHom cghm 19127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-ghm 19128 |
This theorem is referenced by: ghmid 19136 ghminv 19137 ghmsub 19138 ghmmhm 19140 ghmrn 19143 resghm 19146 ghmpreima 19152 ghmnsgima 19154 ghmnsgpreima 19155 ghmf1o 19162 lactghmga 19314 invghm 19742 ghmplusg 19755 rhmopp 20400 srngadd 20608 islmhm2 20793 cygznlem3 21344 psgnco 21355 evpmodpmf1o 21368 ipdir 21411 evlslem1 21864 mpfind 21889 evl1addd 22080 mdetralt 22330 cpmatacl 22438 mat2pmatghm 22452 ghmcnp 23839 ply1rem 25905 dchrptlem2 26992 abliso 32452 ghmquskerlem1 32790 ghmquskerlem3 32793 rhmpreimaidl 32799 rhmimaidl 32812 r1pquslmic 32944 dimkerim 32988 qqhghm 33254 qqhrhm 33255 fldhmf1 41261 evlsaddval 41442 evladdval 41449 gicabl 42143 |
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