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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 17893 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 17893 | . 2 ⊢ (𝜑 → 𝐻 ∈ Grp) |
9 | fof 6353 | . . 3 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) | |
10 | 7, 9 | syl 17 | . 2 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
11 | 6 | 3expb 1153 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
12 | 1, 2, 3, 4, 5, 8, 10, 11 | isghmd 18020 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ⟶wf 6119 –onto→wfo 6121 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 +gcplusg 16305 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 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 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 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 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-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-ghm 18009 |
This theorem is referenced by: (None) |
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