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Mirrors > Home > MPE Home > Th. List > mhmf | Structured version Visualization version GIF version |
Description: A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
mhmf.b | ⊢ 𝐵 = (Base‘𝑆) |
mhmf.c | ⊢ 𝐶 = (Base‘𝑇) |
Ref | Expression |
---|---|
mhmf | ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhmf.b | . . . 4 ⊢ 𝐵 = (Base‘𝑆) | |
2 | mhmf.c | . . . 4 ⊢ 𝐶 = (Base‘𝑇) | |
3 | eqid 2735 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2735 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | eqid 2735 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
6 | eqid 2735 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | ismhm 18811 | . . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇)))) |
8 | 7 | simprbi 496 | . 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))) |
9 | 8 | simp1d 1141 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 Mndcmnd 18760 MndHom cmhm 18807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-mhm 18809 |
This theorem is referenced by: mhmf1o 18822 mhmvlin 18827 resmhm 18846 resmhm2 18847 resmhm2b 18848 mhmco 18849 mhmimalem 18850 mhmima 18851 mhmeql 18852 pwsco2mhm 18859 gsumwmhm 18871 frmdup3lem 18892 frmdup3 18893 mhmmulg 19146 ghmmhmb 19258 cntzmhm 19372 cntzmhm2 19373 frgpup3lem 19810 gsumzmhm 19970 gsummhm2 19972 gsummptmhm 19973 rhmimasubrnglem 20582 mhmcompl 22400 mdetleib2 22610 mdetf 22617 mdetdiaglem 22620 mdetrlin 22624 mdetrsca 22625 mdetralt 22630 mdetunilem7 22640 mdetunilem8 22641 dchrelbas2 27296 dchrn0 27309 mhmhmeotmd 33888 mhmcopsr 42536 |
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