![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2825 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2825 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | eqid 2825 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
6 | eqid 2825 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | ismhm 17690 | . . 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 492 | . 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 1176 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 +gcplusg 16305 0gc0g 16453 Mndcmnd 17647 MndHom cmhm 17686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-map 8124 df-mhm 17688 |
This theorem is referenced by: mhmf1o 17698 resmhm 17712 resmhm2 17713 resmhm2b 17714 mhmco 17715 mhmima 17716 mhmeql 17717 pwsco2mhm 17724 gsumwmhm 17736 frmdup3lem 17757 frmdup3 17758 mhmmulg 17934 ghmmhmb 18022 cntzmhm 18121 cntzmhm2 18122 frgpup3lem 18543 gsumzmhm 18690 gsummhm2 18692 gsummptmhm 18693 mhmvlin 20570 mdetleib2 20762 mdetf 20769 mdetdiaglem 20772 mdetrlin 20776 mdetrsca 20777 mdetralt 20782 mdetunilem7 20792 mdetunilem8 20793 dchrelbas2 25375 dchrn0 25388 mhmhmeotmd 30507 |
Copyright terms: Public domain | W3C validator |