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Mirrors > Home > MPE Home > Th. List > mhmlin | Structured version Visualization version GIF version |
Description: A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
mhmlin.b | ⊢ 𝐵 = (Base‘𝑆) |
mhmlin.p | ⊢ + = (+g‘𝑆) |
mhmlin.q | ⊢ ⨣ = (+g‘𝑇) |
Ref | Expression |
---|---|
mhmlin | ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhmlin.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
2 | eqid 2733 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | mhmlin.p | . . . . . 6 ⊢ + = (+g‘𝑆) | |
4 | mhmlin.q | . . . . . 6 ⊢ ⨣ = (+g‘𝑇) | |
5 | eqid 2733 | . . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
6 | eqid 2733 | . . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | ismhm 18673 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇)))) |
8 | 7 | simprbi 498 | . . . 4 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) |
9 | 8 | simp2d 1144 | . . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
10 | fvoveq1 7432 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦))) | |
11 | fveq2 6892 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
12 | 11 | oveq1d 7424 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦))) |
13 | 10, 12 | eqeq12d 2749 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)))) |
14 | oveq2 7417 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌)) | |
15 | 14 | fveq2d 6896 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌))) |
16 | fveq2 6892 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌)) | |
17 | 16 | oveq2d 7425 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))) |
18 | 15, 17 | eqeq12d 2749 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))) |
19 | 13, 18 | rspc2v 3623 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))) |
20 | 9, 19 | syl5com 31 | . 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))) |
21 | 20 | 3impib 1117 | 1 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 0gc0g 17385 Mndcmnd 18625 MndHom cmhm 18669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-map 8822 df-mhm 18671 |
This theorem is referenced by: mhmf1o 18682 resmhm 18701 resmhm2 18702 resmhm2b 18703 mhmco 18704 mhmimalem 18705 mhmeql 18707 pwsco2mhm 18714 gsumwmhm 18726 mhmmulg 18995 ghmmhmb 19103 cntzmhm 19205 gsumzmhm 19805 rhmmul 20264 evlslem1 21645 mpfind 21670 mhmvlin 21899 mdetunilem7 22120 dchrzrhmul 26749 dchrmulcl 26752 dchrn0 26753 dchrinvcl 26756 dchrsum2 26771 sum2dchr 26777 mhmhmeotmd 32907 rhmimasubrnglem 46744 |
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