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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 8230 | . . . 4 ⊢ (𝑧 ∈ 2o → (1o ∖ 𝑧) ∈ 2o) | |
2 | opelxpi 5588 | . . . 4 ⊢ ((𝑦 ∈ 𝐼 ∧ (1o ∖ 𝑧) ∈ 2o) → 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o)) | |
3 | 1, 2 | sylan2 596 | . . 3 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2o) → 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o)) |
4 | 3 | rgen2 3124 | . 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 7838 | . 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 〈𝑦, (1o ∖ 𝑧)〉 ∈ (𝐼 × 2o) ↔ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) |
7 | 4, 6 | mpbi 233 | 1 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∖ cdif 3863 〈cop 4547 × cxp 5549 ⟶wf 6376 ∈ cmpo 7215 1oc1o 8195 2oc2o 8196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ord 6216 df-on 6217 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-1o 8202 df-2o 8203 |
This theorem is referenced by: efgtf 19112 efgtlen 19116 efginvrel2 19117 efginvrel1 19118 efgredleme 19133 efgredlemc 19135 efgcpbllemb 19145 frgp0 19150 frgpinv 19154 vrgpinv 19159 frgpnabllem1 19258 |
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