![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ismhmd | Structured version Visualization version GIF version |
Description: Deduction version of ismhm 18608. (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 3194 | . . 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 1129 | . 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 18608 | . 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 1345 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 MndHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 0gc0g 17326 Mndcmnd 18561 MndHom cmhm 18604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8770 df-mhm 18606 |
This theorem is referenced by: pwspjmhmmgpd 20048 mhphflem 40813 |
Copyright terms: Public domain | W3C validator |