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Mirrors > Home > MPE Home > Th. List > grpodivval | Structured version Visualization version GIF version |
Description: Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpdiv.1 | ⊢ 𝑋 = ran 𝐺 |
grpdiv.2 | ⊢ 𝑁 = (inv‘𝐺) |
grpdiv.3 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) |
Ref | Expression |
---|---|
grpodivval | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpdiv.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
2 | grpdiv.2 | . . . . 5 ⊢ 𝑁 = (inv‘𝐺) | |
3 | grpdiv.3 | . . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
4 | 1, 2, 3 | grpodivfval 28311 | . . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))) |
5 | 4 | oveqd 7173 | . . 3 ⊢ (𝐺 ∈ GrpOp → (𝐴𝐷𝐵) = (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))𝐵)) |
6 | oveq1 7163 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐺(𝑁‘𝑦)) = (𝐴𝐺(𝑁‘𝑦))) | |
7 | fveq2 6670 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵)) | |
8 | 7 | oveq2d 7172 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(𝑁‘𝑦)) = (𝐴𝐺(𝑁‘𝐵))) |
9 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦))) | |
10 | ovex 7189 | . . . 4 ⊢ (𝐴𝐺(𝑁‘𝐵)) ∈ V | |
11 | 6, 8, 9, 10 | ovmpo 7310 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(𝑁‘𝑦)))𝐵) = (𝐴𝐺(𝑁‘𝐵))) |
12 | 5, 11 | sylan9eq 2876 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵))) |
13 | 12 | 3impb 1111 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 GrpOpcgr 28266 invcgn 28268 /𝑔 cgs 28269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-gdiv 28273 |
This theorem is referenced by: grpodivinv 28313 grpoinvdiv 28314 grpodivdiv 28317 grpomuldivass 28318 grpodivid 28319 grponpcan 28320 ablodivdiv4 28331 nvmval 28419 rngosub 35223 |
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