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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 4878 | . 2 ⊢ (𝑎 = 𝐴 → 〈𝑎, (1o ∖ 𝑏)〉 = 〈𝐴, (1o ∖ 𝑏)〉) | |
2 | difeq2 4130 | . . 3 ⊢ (𝑏 = 𝐵 → (1o ∖ 𝑏) = (1o ∖ 𝐵)) | |
3 | 2 | opeq2d 4885 | . 2 ⊢ (𝑏 = 𝐵 → 〈𝐴, (1o ∖ 𝑏)〉 = 〈𝐴, (1o ∖ 𝐵)〉) |
4 | efgmval.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
5 | opeq1 4878 | . . . 4 ⊢ (𝑦 = 𝑎 → 〈𝑦, (1o ∖ 𝑧)〉 = 〈𝑎, (1o ∖ 𝑧)〉) | |
6 | difeq2 4130 | . . . . 5 ⊢ (𝑧 = 𝑏 → (1o ∖ 𝑧) = (1o ∖ 𝑏)) | |
7 | 6 | opeq2d 4885 | . . . 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 2763 | . 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ 〈𝑎, (1o ∖ 𝑏)〉) |
10 | opex 5475 | . 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 1537 ∈ wcel 2106 ∖ cdif 3960 〈cop 4637 (class class class)co 7431 ∈ cmpo 7433 1oc1o 8498 2oc2o 8499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: efgmnvl 19747 efgval2 19757 vrgpinv 19802 frgpuptinv 19804 frgpuplem 19805 frgpnabllem1 19906 |
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