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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 4796 | . 2 ⊢ (𝑎 = 𝐴 → 〈𝑎, (1o ∖ 𝑏)〉 = 〈𝐴, (1o ∖ 𝑏)〉) | |
2 | difeq2 4086 | . . 3 ⊢ (𝑏 = 𝐵 → (1o ∖ 𝑏) = (1o ∖ 𝐵)) | |
3 | 2 | opeq2d 4803 | . 2 ⊢ (𝑏 = 𝐵 → 〈𝐴, (1o ∖ 𝑏)〉 = 〈𝐴, (1o ∖ 𝐵)〉) |
4 | efgmval.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
5 | opeq1 4796 | . . . 4 ⊢ (𝑦 = 𝑎 → 〈𝑦, (1o ∖ 𝑧)〉 = 〈𝑎, (1o ∖ 𝑧)〉) | |
6 | difeq2 4086 | . . . . 5 ⊢ (𝑧 = 𝑏 → (1o ∖ 𝑧) = (1o ∖ 𝑏)) | |
7 | 6 | opeq2d 4803 | . . . 4 ⊢ (𝑧 = 𝑏 → 〈𝑎, (1o ∖ 𝑧)〉 = 〈𝑎, (1o ∖ 𝑏)〉) |
8 | 5, 7 | cbvmpov 7242 | . . 3 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ 〈𝑎, (1o ∖ 𝑏)〉) |
9 | 4, 8 | eqtri 2843 | . 2 ⊢ 𝑀 = (𝑎 ∈ 𝐼, 𝑏 ∈ 2o ↦ 〈𝑎, (1o ∖ 𝑏)〉) |
10 | opex 5349 | . 2 ⊢ 〈𝐴, (1o ∖ 𝐵)〉 ∈ V | |
11 | 1, 3, 9, 10 | ovmpo 7303 | 1 ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = 〈𝐴, (1o ∖ 𝐵)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∖ cdif 3926 〈cop 4566 (class class class)co 7149 ∈ cmpo 7151 1oc1o 8088 2oc2o 8089 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 |
This theorem is referenced by: efgmnvl 18835 efgval2 18845 vrgpinv 18890 frgpuptinv 18892 frgpuplem 18893 frgpnabllem1 18988 |
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