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Mirrors > Home > MPE Home > Th. List > ismhmd | Structured version Visualization version GIF version |
Description: Deduction version of ismhm 18711. (Contributed by SN, 27-Jul-2024.) |
Ref | Expression |
---|---|
ismhmd.b | ⊢ 𝐵 = (Base‘𝑆) |
ismhmd.c | ⊢ 𝐶 = (Base‘𝑇) |
ismhmd.p | ⊢ + = (+g‘𝑆) |
ismhmd.q | ⊢ ⨣ = (+g‘𝑇) |
ismhmd.0 | ⊢ 0 = (0g‘𝑆) |
ismhmd.z | ⊢ 𝑍 = (0g‘𝑇) |
ismhmd.s | ⊢ (𝜑 → 𝑆 ∈ Mnd) |
ismhmd.t | ⊢ (𝜑 → 𝑇 ∈ Mnd) |
ismhmd.f | ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
ismhmd.a | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
ismhmd.h | ⊢ (𝜑 → (𝐹‘ 0 ) = 𝑍) |
Ref | Expression |
---|---|
ismhmd | ⊢ (𝜑 → 𝐹 ∈ (𝑆 MndHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismhmd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ Mnd) | |
2 | ismhmd.t | . 2 ⊢ (𝜑 → 𝑇 ∈ Mnd) | |
3 | ismhmd.f | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | |
4 | ismhmd.a | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) | |
5 | 4 | ralrimivva 3192 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
6 | ismhmd.h | . . 3 ⊢ (𝜑 → (𝐹‘ 0 ) = 𝑍) | |
7 | 3, 5, 6 | 3jca 1125 | . 2 ⊢ (𝜑 → (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑍)) |
8 | ismhmd.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
9 | ismhmd.c | . . 3 ⊢ 𝐶 = (Base‘𝑇) | |
10 | ismhmd.p | . . 3 ⊢ + = (+g‘𝑆) | |
11 | ismhmd.q | . . 3 ⊢ ⨣ = (+g‘𝑇) | |
12 | ismhmd.0 | . . 3 ⊢ 0 = (0g‘𝑆) | |
13 | ismhmd.z | . . 3 ⊢ 𝑍 = (0g‘𝑇) | |
14 | 8, 9, 10, 11, 12, 13 | ismhm 18711 | . 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 ) = 𝑍))) |
15 | 1, 2, 7, 14 | syl21anbrc 1341 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 MndHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 Basecbs 17149 +gcplusg 17202 0gc0g 17390 Mndcmnd 18663 MndHom cmhm 18707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-map 8819 df-mhm 18709 |
This theorem is referenced by: pwspjmhmmgpd 20223 imasmhm 32962 mhphflem 41699 |
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