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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 8449 | . . . 4 ⊢ (𝑧 ∈ 2o → (1o ∖ 𝑧) ∈ 2o) | |
2 | opelxpi 5670 | . . . 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 3194 | . 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 8000 | . 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o) ↔ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) |
7 | 4, 6 | mpbi 229 | 1 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∖ cdif 3907 〈cop 4592 × cxp 5631 ⟶wf 6492 ∈ cmpo 7359 1oc1o 8405 2oc2o 8406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-ord 6320 df-on 6321 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-1o 8412 df-2o 8413 |
This theorem is referenced by: efgtf 19504 efgtlen 19508 efginvrel2 19509 efginvrel1 19510 efgredleme 19525 efgredlemc 19527 efgcpbllemb 19537 frgp0 19542 frgpinv 19546 vrgpinv 19551 frgpnabllem1 19651 |
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