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Mirrors > Home > MPE Home > Th. List > efgmf | Structured version Visualization version GIF version |
Description: The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
efgmval.m | ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) |
Ref | Expression |
---|---|
efgmf | ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2oconcl 8540 | . . . 4 ⊢ (𝑧 ∈ 2o → (1o ∖ 𝑧) ∈ 2o) | |
2 | opelxpi 5726 | . . . 4 ⊢ ((𝑦 ∈ 𝐼 ∧ (1o ∖ 𝑧) ∈ 2o) → 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o)) | |
3 | 1, 2 | sylan2 593 | . . 3 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o) → 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o)) |
4 | 3 | rgen2 3197 | . 2 ⊢ ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o) |
5 | efgmval.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
6 | 5 | fmpo 8092 | . 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o) ↔ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) |
7 | 4, 6 | mpbi 230 | 1 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∖ cdif 3960 〈cop 4637 × cxp 5687 ⟶wf 6559 ∈ 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 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-1o 8505 df-2o 8506 |
This theorem is referenced by: efgtf 19755 efgtlen 19759 efginvrel2 19760 efginvrel1 19761 efgredleme 19776 efgredlemc 19778 efgcpbllemb 19788 frgp0 19793 frgpinv 19797 vrgpinv 19802 frgpnabllem1 19906 |
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