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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 2818 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | ghmlin.a | . . . . . 6 ⊢ + = (+g‘𝑆) | |
4 | ghmlin.b | . . . . . 6 ⊢ ⨣ = (+g‘𝑇) | |
5 | 1, 2, 3, 4 | isghm 18296 | . . . . 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 7168 | . . . . 5 ⊢ (𝑎 = 𝑈 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑈 + 𝑏))) | |
9 | fveq2 6663 | . . . . . 6 ⊢ (𝑎 = 𝑈 → (𝐹‘𝑎) = (𝐹‘𝑈)) | |
10 | 9 | oveq1d 7160 | . . . . 5 ⊢ (𝑎 = 𝑈 → ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏))) |
11 | 8, 10 | eqeq12d 2834 | . . . 4 ⊢ (𝑎 = 𝑈 → ((𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) ↔ (𝐹‘(𝑈 + 𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)))) |
12 | oveq2 7153 | . . . . . 6 ⊢ (𝑏 = 𝑉 → (𝑈 + 𝑏) = (𝑈 + 𝑉)) | |
13 | 12 | fveq2d 6667 | . . . . 5 ⊢ (𝑏 = 𝑉 → (𝐹‘(𝑈 + 𝑏)) = (𝐹‘(𝑈 + 𝑉))) |
14 | fveq2 6663 | . . . . . 6 ⊢ (𝑏 = 𝑉 → (𝐹‘𝑏) = (𝐹‘𝑉)) | |
15 | 14 | oveq2d 7161 | . . . . 5 ⊢ (𝑏 = 𝑉 → ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))) |
16 | 13, 15 | eqeq12d 2834 | . . . 4 ⊢ (𝑏 = 𝑉 → ((𝐹‘(𝑈 + 𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)) ↔ (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉)))) |
17 | 11, 16 | rspc2v 3630 | . . 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 1107 | 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 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Grpcgrp 18041 GrpHom cghm 18293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-ghm 18294 |
This theorem is referenced by: ghmid 18302 ghminv 18303 ghmsub 18304 ghmmhm 18306 ghmrn 18309 resghm 18312 ghmpreima 18318 ghmnsgima 18320 ghmnsgpreima 18321 ghmf1o 18326 lactghmga 18462 invghm 18883 ghmplusg 18895 srngadd 19557 islmhm2 19739 evlslem1 20223 mpfind 20248 evl1addd 20432 cygznlem3 20644 psgnco 20655 evpmodpmf1o 20668 ipdir 20711 mdetralt 21145 cpmatacl 21252 mat2pmatghm 21266 ghmcnp 22650 ply1rem 24684 dchrptlem2 25768 abliso 30610 rhmopp 30819 dimkerim 30922 qqhghm 31128 qqhrhm 31129 gicabl 39577 |
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