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Mirrors > Home > MPE Home > Th. List > grpoinvdiv | Structured version Visualization version GIF version |
Description: Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpdiv.1 | ⊢ 𝑋 = ran 𝐺 |
grpdiv.2 | ⊢ 𝑁 = (inv‘𝐺) |
grpdiv.3 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) |
Ref | Expression |
---|---|
grpoinvdiv | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpdiv.1 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
2 | grpdiv.2 | . . . 4 ⊢ 𝑁 = (inv‘𝐺) | |
3 | grpdiv.3 | . . . 4 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
4 | 1, 2, 3 | grpodivval 26773 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁‘𝐵))) |
5 | 4 | fveq2d 6107 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝑁‘(𝐴𝐺(𝑁‘𝐵)))) |
6 | 1, 2 | grpoinvcl 26762 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋) |
7 | 6 | 3adant2 1073 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ 𝑋) |
8 | 1, 2 | grpoinvop 26771 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ (𝑁‘𝐵) ∈ 𝑋) → (𝑁‘(𝐴𝐺(𝑁‘𝐵))) = ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴))) |
9 | 7, 8 | syld3an3 1363 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(𝑁‘𝐵))) = ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴))) |
10 | 1, 2 | grpo2inv 26769 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝑁‘𝐵)) = 𝐵) |
11 | 10 | 3adant2 1073 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝑁‘𝐵)) = 𝐵) |
12 | 11 | oveq1d 6564 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐺(𝑁‘𝐴))) |
13 | 1, 2, 3 | grpodivval 26773 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁‘𝐴))) |
14 | 13 | 3com23 1263 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐵𝐺(𝑁‘𝐴))) |
15 | 12, 14 | eqtr4d 2647 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝑁‘𝐵))𝐺(𝑁‘𝐴)) = (𝐵𝐷𝐴)) |
16 | 5, 9, 15 | 3eqtrd 2648 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ran crn 5039 ‘cfv 5804 (class class class)co 6549 GrpOpcgr 26727 invcgn 26729 /𝑔 cgs 26730 |
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-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-1st 7059 df-2nd 7060 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 |
This theorem is referenced by: grpodivdiv 26778 |
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