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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 8131 | . . . 4 ⊢ (𝑧 ∈ 2o → (1o ∖ 𝑧) ∈ 2o) | |
2 | opelxpi 5595 | . . . 4 ⊢ ((𝑦 ∈ 𝐼 ∧ (1o ∖ 𝑧) ∈ 2o) → 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o)) | |
3 | 1, 2 | sylan2 594 | . . 3 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o) → 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o)) |
4 | 3 | rgen2 3206 | . 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 7769 | . 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o) ↔ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) |
7 | 4, 6 | mpbi 232 | 1 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∖ cdif 3936 〈cop 4576 × cxp 5556 ⟶wf 6354 ∈ cmpo 7161 1oc1o 8098 2oc2o 8099 |
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 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-ord 6197 df-on 6198 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-1o 8105 df-2o 8106 |
This theorem is referenced by: efgtf 18851 efgtlen 18855 efginvrel2 18856 efginvrel1 18857 efgredleme 18872 efgredlemc 18874 efgcpbllemb 18884 frgp0 18889 frgpinv 18893 vrgpinv 18898 frgpnabllem1 18996 |
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