| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ghmpropd | Structured version Visualization version GIF version | ||
| Description: Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.) |
| Ref | Expression |
|---|---|
| ghmpropd.a | ⊢ (𝜑 → 𝐵 = (Base‘𝐽)) |
| ghmpropd.b | ⊢ (𝜑 → 𝐶 = (Base‘𝐾)) |
| ghmpropd.c | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
| ghmpropd.d | ⊢ (𝜑 → 𝐶 = (Base‘𝑀)) |
| ghmpropd.e | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
| ghmpropd.f | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦)) |
| Ref | Expression |
|---|---|
| ghmpropd | ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ghmpropd.a | . . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐽)) | |
| 2 | ghmpropd.c | . . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
| 3 | ghmpropd.e | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
| 4 | 1, 2, 3 | grppropd 18969 | . . . . 5 ⊢ (𝜑 → (𝐽 ∈ Grp ↔ 𝐿 ∈ Grp)) |
| 5 | ghmpropd.b | . . . . . 6 ⊢ (𝜑 → 𝐶 = (Base‘𝐾)) | |
| 6 | ghmpropd.d | . . . . . 6 ⊢ (𝜑 → 𝐶 = (Base‘𝑀)) | |
| 7 | ghmpropd.f | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦)) | |
| 8 | 5, 6, 7 | grppropd 18969 | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝑀 ∈ Grp)) |
| 9 | 4, 8 | anbi12d 632 | . . . 4 ⊢ (𝜑 → ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ↔ (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp))) |
| 10 | 1, 5, 2, 6, 3, 7 | mhmpropd 18805 | . . . . 5 ⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀)) |
| 11 | 10 | eleq2d 2827 | . . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀))) |
| 12 | 9, 11 | anbi12d 632 | . . 3 ⊢ (𝜑 → (((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾)) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀)))) |
| 13 | ghmgrp1 19236 | . . . . 5 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐽 ∈ Grp) | |
| 14 | ghmgrp2 19237 | . . . . 5 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐾 ∈ Grp) | |
| 15 | 13, 14 | jca 511 | . . . 4 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → (𝐽 ∈ Grp ∧ 𝐾 ∈ Grp)) |
| 16 | ghmmhmb 19245 | . . . . 5 ⊢ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝐽 GrpHom 𝐾) = (𝐽 MndHom 𝐾)) | |
| 17 | 16 | eleq2d 2827 | . . . 4 ⊢ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐽 MndHom 𝐾))) |
| 18 | 15, 17 | biadanii 822 | . . 3 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾))) |
| 19 | ghmgrp1 19236 | . . . . 5 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝐿 ∈ Grp) | |
| 20 | ghmgrp2 19237 | . . . . 5 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝑀 ∈ Grp) | |
| 21 | 19, 20 | jca 511 | . . . 4 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp)) |
| 22 | ghmmhmb 19245 | . . . . 5 ⊢ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐿 GrpHom 𝑀) = (𝐿 MndHom 𝑀)) | |
| 23 | 22 | eleq2d 2827 | . . . 4 ⊢ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀))) |
| 24 | 21, 23 | biadanii 822 | . . 3 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀))) |
| 25 | 12, 18, 24 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀))) |
| 26 | 25 | eqrdv 2735 | 1 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 MndHom cmhm 18794 Grpcgrp 18951 GrpHom cghm 19230 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-ghm 19231 |
| This theorem is referenced by: rhmpropd 20609 lmhmpropd 21072 evls1maplmhm 22381 |
| Copyright terms: Public domain | W3C validator |