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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 2731 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | ghmlin.a | . . . . . 6 ⊢ + = (+g‘𝑆) | |
4 | ghmlin.b | . . . . . 6 ⊢ ⨣ = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | isghm 19137 | . . . . 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 7435 | . . . . 5 ⊢ (𝑎 = 𝑈 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑈 + 𝑏))) | |
9 | fveq2 6891 | . . . . . 6 ⊢ (𝑎 = 𝑈 → (𝐹‘𝑎) = (𝐹‘𝑈)) | |
10 | 9 | oveq1d 7427 | . . . . 5 ⊢ (𝑎 = 𝑈 → ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏))) |
11 | 8, 10 | eqeq12d 2747 | . . . 4 ⊢ (𝑎 = 𝑈 → ((𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) ↔ (𝐹‘(𝑈 + 𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)))) |
12 | oveq2 7420 | . . . . . 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 7428 | . . . . 5 ⊢ (𝑏 = 𝑉 → ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))) |
16 | 13, 15 | eqeq12d 2747 | . . . 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 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 1540 ∈ wcel 2105 ∀wral 3060 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 Grpcgrp 18861 GrpHom cghm 19134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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 7415 df-oprab 7416 df-mpo 7417 df-ghm 19135 |
This theorem is referenced by: ghmid 19143 ghminv 19144 ghmsub 19145 ghmmhm 19147 ghmrn 19150 resghm 19153 ghmpreima 19159 ghmnsgima 19161 ghmnsgpreima 19162 ghmf1o 19169 lactghmga 19321 invghm 19749 ghmplusg 19762 rhmopp 20407 srngadd 20696 islmhm2 20882 cygznlem3 21435 psgnco 21446 evpmodpmf1o 21459 ipdir 21502 evlslem1 21956 mpfind 21981 evl1addd 22180 mdetralt 22430 cpmatacl 22538 mat2pmatghm 22552 ghmcnp 23939 ply1rem 26019 dchrptlem2 27112 abliso 32631 ghmquskerlem1 32969 ghmquskerlem3 32972 rhmpreimaidl 32978 rhmimaidl 32991 r1pquslmic 33123 dimkerim 33167 qqhghm 33433 qqhrhm 33434 fldhmf1 41424 evlsaddval 41605 evladdval 41612 gicabl 42306 |
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