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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 18441 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 18441 | . 2 ⊢ (𝜑 → 𝐻 ∈ Grp) |
9 | fof 6611 | . . 3 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) | |
10 | 7, 9 | syl 17 | . 2 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
11 | 6 | 3expb 1122 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
12 | 1, 2, 3, 4, 5, 8, 10, 11 | isghmd 18585 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 Grpcgrp 18319 GrpHom cghm 18573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-minusg 18323 df-ghm 18574 |
This theorem is referenced by: (None) |
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