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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 17791 | . . . . 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 17791 | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝑀 ∈ Grp)) |
9 | 4, 8 | anbi12d 626 | . . . 4 ⊢ (𝜑 → ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ↔ (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp))) |
10 | 1, 5, 2, 6, 3, 7 | mhmpropd 17694 | . . . . 5 ⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀)) |
11 | 10 | eleq2d 2892 | . . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀))) |
12 | 9, 11 | anbi12d 626 | . . 3 ⊢ (𝜑 → (((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾)) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀)))) |
13 | ghmgrp1 18013 | . . . . 5 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐽 ∈ Grp) | |
14 | ghmgrp2 18014 | . . . . 5 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐾 ∈ Grp) | |
15 | 13, 14 | jca 509 | . . . 4 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → (𝐽 ∈ Grp ∧ 𝐾 ∈ Grp)) |
16 | ghmmhmb 18022 | . . . . 5 ⊢ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝐽 GrpHom 𝐾) = (𝐽 MndHom 𝐾)) | |
17 | 16 | eleq2d 2892 | . . . 4 ⊢ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐽 MndHom 𝐾))) |
18 | 15, 17 | biadanii 859 | . . 3 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾))) |
19 | ghmgrp1 18013 | . . . . 5 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝐿 ∈ Grp) | |
20 | ghmgrp2 18014 | . . . . 5 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝑀 ∈ Grp) | |
21 | 19, 20 | jca 509 | . . . 4 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp)) |
22 | ghmmhmb 18022 | . . . . 5 ⊢ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐿 GrpHom 𝑀) = (𝐿 MndHom 𝑀)) | |
23 | 22 | eleq2d 2892 | . . . 4 ⊢ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀))) |
24 | 21, 23 | biadanii 859 | . . 3 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀))) |
25 | 12, 18, 24 | 3bitr4g 306 | . 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀))) |
26 | 25 | eqrdv 2823 | 1 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 +gcplusg 16305 MndHom cmhm 17686 Grpcgrp 17776 GrpHom cghm 18008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-map 8124 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-grp 17779 df-ghm 18009 |
This theorem is referenced by: rhmpropd 19171 lmhmpropd 19432 |
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