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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 4763 | . 2 ⊢ (𝑎 = 𝐴 → 〈𝑎, (1o ∖ 𝑏)〉 = 〈𝐴, (1o ∖ 𝑏)〉) | |
2 | difeq2 4044 | . . 3 ⊢ (𝑏 = 𝐵 → (1o ∖ 𝑏) = (1o ∖ 𝐵)) | |
3 | 2 | opeq2d 4772 | . 2 ⊢ (𝑏 = 𝐵 → 〈𝐴, (1o ∖ 𝑏)〉 = 〈𝐴, (1o ∖ 𝐵)〉) |
4 | efgmval.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
5 | opeq1 4763 | . . . 4 ⊢ (𝑦 = 𝑎 → 〈𝑦, (1o ∖ 𝑧)〉 = 〈𝑎, (1o ∖ 𝑧)〉) | |
6 | difeq2 4044 | . . . . 5 ⊢ (𝑧 = 𝑏 → (1o ∖ 𝑧) = (1o ∖ 𝑏)) | |
7 | 6 | opeq2d 4772 | . . . 4 ⊢ (𝑧 = 𝑏 → 〈𝑎, (1o ∖ 𝑧)〉 = 〈𝑎, (1o ∖ 𝑏)〉) |
8 | 5, 7 | cbvmpov 7228 | . . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ 〈𝑎, (1o ∖ 𝑏)〉) |
9 | 4, 8 | eqtri 2821 | . 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ 〈𝑎, (1o ∖ 𝑏)〉) |
10 | opex 5321 | . 2 ⊢ 〈𝐴, (1o ∖ 𝐵)〉 ∈ V | |
11 | 1, 3, 9, 10 | ovmpo 7289 | 1 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = 〈𝐴, (1o ∖ 𝐵)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 〈cop 4531 (class class class)co 7135 ∈ cmpo 7137 1oc1o 8078 2oc2o 8079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: efgmnvl 18832 efgval2 18842 vrgpinv 18887 frgpuptinv 18889 frgpuplem 18890 frgpnabllem1 18986 |
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