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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 2732 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2732 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | eqid 2732 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
6 | eqid 2732 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | ismhm 18669 | . . 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 497 | . 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 1142 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:𝐵⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 +gcplusg 17193 0gc0g 17381 Mndcmnd 18621 MndHom cmhm 18665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8818 df-mhm 18667 |
This theorem is referenced by: mhmf1o 18678 resmhm 18697 resmhm2 18698 resmhm2b 18699 mhmco 18700 mhmimalem 18701 mhmima 18702 mhmeql 18703 pwsco2mhm 18710 gsumwmhm 18722 frmdup3lem 18743 frmdup3 18744 mhmmulg 18989 ghmmhmb 19097 cntzmhm 19199 cntzmhm2 19200 frgpup3lem 19639 gsumzmhm 19799 gsummhm2 19801 gsummptmhm 19802 mhmvlin 21890 mdetleib2 22081 mdetf 22088 mdetdiaglem 22091 mdetrlin 22095 mdetrsca 22096 mdetralt 22101 mdetunilem7 22111 mdetunilem8 22112 dchrelbas2 26729 dchrn0 26742 mhmhmeotmd 32895 mhmcompl 41117 rhmimasubrnglem 46728 |
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