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Mirrors > Home > ILE Home > Th. List > grpsubpropdg | GIF version |
Description: Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.) |
Ref | Expression |
---|---|
grpsubpropd.b | ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) |
grpsubpropd.p | ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) |
grpsubpropdg.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
grpsubpropdg.h | ⊢ (𝜑 → 𝐻 ∈ 𝑊) |
Ref | Expression |
---|---|
grpsubpropdg | ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubpropd.b | . . 3 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) | |
2 | grpsubpropd.p | . . . 4 ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) | |
3 | eqidd 2178 | . . . 4 ⊢ (𝜑 → 𝑎 = 𝑎) | |
4 | eqidd 2178 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | grpsubpropdg.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
6 | grpsubpropdg.h | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ 𝑊) | |
7 | 2 | oveqdr 5905 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦)) |
8 | 4, 1, 5, 6, 7 | grpinvpropdg 12950 | . . . . 5 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻)) |
9 | 8 | fveq1d 5519 | . . . 4 ⊢ (𝜑 → ((invg‘𝐺)‘𝑏) = ((invg‘𝐻)‘𝑏)) |
10 | 2, 3, 9 | oveq123d 5898 | . . 3 ⊢ (𝜑 → (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)) = (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏))) |
11 | 1, 1, 10 | mpoeq123dv 5939 | . 2 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏))) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))) |
12 | eqid 2177 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
13 | eqid 2177 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
14 | eqid 2177 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
15 | eqid 2177 | . . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
16 | 12, 13, 14, 15 | grpsubfvalg 12923 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)))) |
17 | 5, 16 | syl 14 | . 2 ⊢ (𝜑 → (-g‘𝐺) = (𝑎 ∈ (Base‘𝐺), 𝑏 ∈ (Base‘𝐺) ↦ (𝑎(+g‘𝐺)((invg‘𝐺)‘𝑏)))) |
18 | eqid 2177 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
19 | eqid 2177 | . . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
20 | eqid 2177 | . . . 4 ⊢ (invg‘𝐻) = (invg‘𝐻) | |
21 | eqid 2177 | . . . 4 ⊢ (-g‘𝐻) = (-g‘𝐻) | |
22 | 18, 19, 20, 21 | grpsubfvalg 12923 | . . 3 ⊢ (𝐻 ∈ 𝑊 → (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))) |
23 | 6, 22 | syl 14 | . 2 ⊢ (𝜑 → (-g‘𝐻) = (𝑎 ∈ (Base‘𝐻), 𝑏 ∈ (Base‘𝐻) ↦ (𝑎(+g‘𝐻)((invg‘𝐻)‘𝑏)))) |
24 | 11, 17, 23 | 3eqtr4d 2220 | 1 ⊢ (𝜑 → (-g‘𝐺) = (-g‘𝐻)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ‘cfv 5218 (class class class)co 5877 ∈ cmpo 5879 Basecbs 12464 +gcplusg 12538 invgcminusg 12883 -gcsg 12884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-inn 8922 df-ndx 12467 df-slot 12468 df-base 12470 df-0g 12712 df-minusg 12886 df-sbg 12887 |
This theorem is referenced by: (None) |
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