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Mirrors > Home > MPE Home > Th. List > ismhmd | Structured version Visualization version GIF version |
Description: Deduction version of ismhm 18742. (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 3197 | . . 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 1126 | . 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 18742 | . 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 1342 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 MndHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 0gc0g 17421 Mndcmnd 18694 MndHom cmhm 18738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-mhm 18740 |
This theorem is referenced by: pwspjmhmmgpd 20264 imasmhm 33079 mhphflem 41829 |
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