Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ismhm0 | Structured version Visualization version GIF version |
Description: Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.) |
Ref | Expression |
---|---|
ismhm0.b | ⊢ 𝐵 = (Base‘𝑆) |
ismhm0.c | ⊢ 𝐶 = (Base‘𝑇) |
ismhm0.p | ⊢ + = (+g‘𝑆) |
ismhm0.q | ⊢ ⨣ = (+g‘𝑇) |
ismhm0.z | ⊢ 0 = (0g‘𝑆) |
ismhm0.y | ⊢ 𝑌 = (0g‘𝑇) |
Ref | Expression |
---|---|
ismhm0 | ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismhm0.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
2 | ismhm0.c | . . 3 ⊢ 𝐶 = (Base‘𝑇) | |
3 | ismhm0.p | . . 3 ⊢ + = (+g‘𝑆) | |
4 | ismhm0.q | . . 3 ⊢ ⨣ = (+g‘𝑇) | |
5 | ismhm0.z | . . 3 ⊢ 0 = (0g‘𝑆) | |
6 | ismhm0.y | . . 3 ⊢ 𝑌 = (0g‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | ismhm 17960 | . 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌))) |
8 | df-3an 1085 | . . . 4 ⊢ ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌) ↔ ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ∧ (𝐹‘ 0 ) = 𝑌)) | |
9 | mndmgm 17920 | . . . . . . . 8 ⊢ (𝑆 ∈ Mnd → 𝑆 ∈ Mgm) | |
10 | mndmgm 17920 | . . . . . . . 8 ⊢ (𝑇 ∈ Mnd → 𝑇 ∈ Mgm) | |
11 | 9, 10 | anim12i 614 | . . . . . . 7 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) |
12 | 11 | biantrurd 535 | . . . . . 6 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))))) |
13 | 1, 2, 3, 4 | ismgmhm 44057 | . . . . . 6 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))) |
14 | 12, 13 | syl6bbr 291 | . . . . 5 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑇))) |
15 | 14 | anbi1d 631 | . . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ∧ (𝐹‘ 0 ) = 𝑌) ↔ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌))) |
16 | 8, 15 | syl5bb 285 | . . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌) ↔ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌))) |
17 | 16 | pm5.32i 577 | . 2 ⊢ (((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌))) |
18 | 7, 17 | bitri 277 | 1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹‘ 0 ) = 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 0gc0g 16715 Mgmcmgm 17852 Mndcmnd 17913 MndHom cmhm 17956 MgmHom cmgmhm 44051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-mgmhm 44053 |
This theorem is referenced by: c0snmhm 44193 |
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