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| Mirrors > Home > MPE Home > Th. List > efgmval | Structured version Visualization version GIF version | ||
| Description: Value of the formal inverse operation for the generating set of a free group. (Contributed by Mario Carneiro, 27-Sep-2015.) |
| Ref | Expression |
|---|---|
| efgmval.m | ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) |
| Ref | Expression |
|---|---|
| efgmval | ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = 〈𝐴, (1o ∖ 𝐵)〉) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opeq1 4873 | . 2 ⊢ (𝑎 = 𝐴 → 〈𝑎, (1o ∖ 𝑏)〉 = 〈𝐴, (1o ∖ 𝑏)〉) | |
| 2 | difeq2 4120 | . . 3 ⊢ (𝑏 = 𝐵 → (1o ∖ 𝑏) = (1o ∖ 𝐵)) | |
| 3 | 2 | opeq2d 4880 | . 2 ⊢ (𝑏 = 𝐵 → 〈𝐴, (1o ∖ 𝑏)〉 = 〈𝐴, (1o ∖ 𝐵)〉) |
| 4 | efgmval.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
| 5 | opeq1 4873 | . . . 4 ⊢ (𝑦 = 𝑎 → 〈𝑦, (1o ∖ 𝑧)〉 = 〈𝑎, (1o ∖ 𝑧)〉) | |
| 6 | difeq2 4120 | . . . . 5 ⊢ (𝑧 = 𝑏 → (1o ∖ 𝑧) = (1o ∖ 𝑏)) | |
| 7 | 6 | opeq2d 4880 | . . . 4 ⊢ (𝑧 = 𝑏 → 〈𝑎, (1o ∖ 𝑧)〉 = 〈𝑎, (1o ∖ 𝑏)〉) |
| 8 | 5, 7 | cbvmpov 7528 | . . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ 〈𝑎, (1o ∖ 𝑏)〉) |
| 9 | 4, 8 | eqtri 2765 | . 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ 〈𝑎, (1o ∖ 𝑏)〉) |
| 10 | opex 5469 | . 2 ⊢ 〈𝐴, (1o ∖ 𝐵)〉 ∈ V | |
| 11 | 1, 3, 9, 10 | ovmpo 7593 | 1 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = 〈𝐴, (1o ∖ 𝐵)〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 〈cop 4632 (class class class)co 7431 ∈ cmpo 7433 1oc1o 8499 2oc2o 8500 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: efgmnvl 19732 efgval2 19742 vrgpinv 19787 frgpuptinv 19789 frgpuplem 19790 frgpnabllem1 19891 |
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