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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 4805 | . 2 ⊢ (𝑎 = 𝐴 → 〈𝑎, (1o ∖ 𝑏)〉 = 〈𝐴, (1o ∖ 𝑏)〉) | |
2 | difeq2 4095 | . . 3 ⊢ (𝑏 = 𝐵 → (1o ∖ 𝑏) = (1o ∖ 𝐵)) | |
3 | 2 | opeq2d 4812 | . 2 ⊢ (𝑏 = 𝐵 → 〈𝐴, (1o ∖ 𝑏)〉 = 〈𝐴, (1o ∖ 𝐵)〉) |
4 | efgmval.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
5 | opeq1 4805 | . . . 4 ⊢ (𝑦 = 𝑎 → 〈𝑦, (1o ∖ 𝑧)〉 = 〈𝑎, (1o ∖ 𝑧)〉) | |
6 | difeq2 4095 | . . . . 5 ⊢ (𝑧 = 𝑏 → (1o ∖ 𝑧) = (1o ∖ 𝑏)) | |
7 | 6 | opeq2d 4812 | . . . 4 ⊢ (𝑧 = 𝑏 → 〈𝑎, (1o ∖ 𝑧)〉 = 〈𝑎, (1o ∖ 𝑏)〉) |
8 | 5, 7 | cbvmpov 7251 | . . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ 〈𝑎, (1o ∖ 𝑏)〉) |
9 | 4, 8 | eqtri 2846 | . 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ 〈𝑎, (1o ∖ 𝑏)〉) |
10 | opex 5358 | . 2 ⊢ 〈𝐴, (1o ∖ 𝐵)〉 ∈ V | |
11 | 1, 3, 9, 10 | ovmpo 7312 | 1 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = 〈𝐴, (1o ∖ 𝐵)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 〈cop 4575 (class class class)co 7158 ∈ cmpo 7160 1oc1o 8097 2oc2o 8098 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 |
This theorem is referenced by: efgmnvl 18842 efgval2 18852 vrgpinv 18897 frgpuptinv 18899 frgpuplem 18900 frgpnabllem1 18995 |
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