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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 18982 | . . . . 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 18982 | . . . . 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 18818 | . . . . 5 ⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀)) |
11 | 10 | eleq2d 2825 | . . . 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 19249 | . . . . 5 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐽 ∈ Grp) | |
14 | ghmgrp2 19250 | . . . . 5 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐾 ∈ Grp) | |
15 | 13, 14 | jca 511 | . . . 4 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → (𝐽 ∈ Grp ∧ 𝐾 ∈ Grp)) |
16 | ghmmhmb 19258 | . . . . 5 ⊢ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝐽 GrpHom 𝐾) = (𝐽 MndHom 𝐾)) | |
17 | 16 | eleq2d 2825 | . . . 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 19249 | . . . . 5 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝐿 ∈ Grp) | |
20 | ghmgrp2 19250 | . . . . 5 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝑀 ∈ Grp) | |
21 | 19, 20 | jca 511 | . . . 4 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp)) |
22 | ghmmhmb 19258 | . . . . 5 ⊢ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐿 GrpHom 𝑀) = (𝐿 MndHom 𝑀)) | |
23 | 22 | eleq2d 2825 | . . . 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 2733 | 1 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 MndHom cmhm 18807 Grpcgrp 18964 GrpHom cghm 19243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-ghm 19244 |
This theorem is referenced by: rhmpropd 20626 lmhmpropd 21090 evls1maplmhm 22397 |
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