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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 2736 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
4 | eqid 2736 | . . . 4 ⊢ (+g‘𝑇) = (+g‘𝑇) | |
5 | eqid 2736 | . . . 4 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
6 | eqid 2736 | . . . 4 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | ismhm 18557 | . . 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 3062 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 Basecbs 17037 +gcplusg 17087 0gc0g 17275 Mndcmnd 18510 MndHom cmhm 18553 |
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 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-map 8725 df-mhm 18555 |
This theorem is referenced by: mhmf1o 18566 resmhm 18585 resmhm2 18586 resmhm2b 18587 mhmco 18588 mhmima 18589 mhmeql 18590 pwsco2mhm 18597 gsumwmhm 18609 frmdup3lem 18630 frmdup3 18631 mhmmulg 18870 ghmmhmb 18972 cntzmhm 19072 cntzmhm2 19073 frgpup3lem 19512 gsumzmhm 19667 gsummhm2 19669 gsummptmhm 19670 mhmvlin 21692 mdetleib2 21883 mdetf 21890 mdetdiaglem 21893 mdetrlin 21897 mdetrsca 21898 mdetralt 21903 mdetunilem7 21913 mdetunilem8 21914 dchrelbas2 26531 dchrn0 26544 mhmhmeotmd 32336 |
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