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Mirrors > Home > MPE Home > Th. List > ghmfghm | Structured version Visualization version GIF version |
Description: The function fulfilling the conditions of ghmgrp 17362 is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.) |
Ref | Expression |
---|---|
ghmabl.x | ⊢ 𝑋 = (Base‘𝐺) |
ghmabl.y | ⊢ 𝑌 = (Base‘𝐻) |
ghmabl.p | ⊢ + = (+g‘𝐺) |
ghmabl.q | ⊢ ⨣ = (+g‘𝐻) |
ghmabl.f | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
ghmabl.1 | ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) |
ghmfghm.3 | ⊢ (𝜑 → 𝐺 ∈ Grp) |
Ref | Expression |
---|---|
ghmfghm | ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmabl.x | . 2 ⊢ 𝑋 = (Base‘𝐺) | |
2 | ghmabl.y | . 2 ⊢ 𝑌 = (Base‘𝐻) | |
3 | ghmabl.p | . 2 ⊢ + = (+g‘𝐺) | |
4 | ghmabl.q | . 2 ⊢ ⨣ = (+g‘𝐻) | |
5 | ghmfghm.3 | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
6 | ghmabl.f | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) | |
7 | ghmabl.1 | . . 3 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) | |
8 | 6, 1, 2, 3, 4, 7, 5 | ghmgrp 17362 | . 2 ⊢ (𝜑 → 𝐻 ∈ Grp) |
9 | fof 6028 | . . 3 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) | |
10 | 7, 9 | syl 17 | . 2 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
11 | 6 | 3expb 1258 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
12 | 1, 2, 3, 4, 5, 8, 10, 11 | isghmd 17492 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Grpcgrp 17245 GrpHom cghm 17480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-ghm 17481 |
This theorem is referenced by: (None) |
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