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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 2818 | . . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | mhmlin.p | . . . . . 6 ⊢ + = (+g‘𝑆) | |
4 | mhmlin.q | . . . . . 6 ⊢ ⨣ = (+g‘𝑇) | |
5 | eqid 2818 | . . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
6 | eqid 2818 | . . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | ismhm 17946 | . . . . 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 497 | . . . 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 1135 | . . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
10 | fvoveq1 7168 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦))) | |
11 | fveq2 6663 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
12 | 11 | oveq1d 7160 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦))) |
13 | 10, 12 | eqeq12d 2834 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)))) |
14 | oveq2 7153 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌)) | |
15 | 14 | fveq2d 6667 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌))) |
16 | fveq2 6663 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌)) | |
17 | 16 | oveq2d 7161 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))) |
18 | 15, 17 | eqeq12d 2834 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))) |
19 | 13, 18 | rspc2v 3630 | . . 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 1108 | 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 Mndcmnd 17899 MndHom cmhm 17942 |
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-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-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 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-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-mhm 17944 |
This theorem is referenced by: mhmf1o 17954 resmhm 17973 resmhm2 17974 resmhm2b 17975 mhmco 17976 mhmima 17977 mhmeql 17978 pwsco2mhm 17985 gsumwmhm 17998 mhmmulg 18206 ghmmhmb 18307 cntzmhm 18407 gsumzmhm 18986 rhmmul 19408 evlslem1 20223 mpfind 20248 mhmvlin 20936 mdetunilem7 21155 dchrzrhmul 25749 dchrmulcl 25752 dchrn0 25753 dchrinvcl 25756 dchrsum2 25771 sum2dchr 25777 mhmhmeotmd 31069 |
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