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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 7583 | . . . 4 ⊢ (𝑧 ∈ 2𝑜 → (1𝑜 ∖ 𝑧) ∈ 2𝑜) | |
2 | opelxpi 5148 | . . . 4 ⊢ ((𝑦 ∈ 𝐼 ∧ (1𝑜 ∖ 𝑧) ∈ 2𝑜) → 〈𝑦, (1𝑜 ∖ 𝑧)〉 ∈ (𝐼 × 2𝑜)) | |
3 | 1, 2 | sylan2 491 | . . 3 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑧 ∈ 2𝑜) → 〈𝑦, (1𝑜 ∖ 𝑧)〉 ∈ (𝐼 × 2𝑜)) |
4 | 3 | rgen2 2975 | . 2 ⊢ ∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 〈𝑦, (1𝑜 ∖ 𝑧)〉 ∈ (𝐼 × 2𝑜) |
5 | efgmval.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 〈𝑦, (1𝑜 ∖ 𝑧)〉) | |
6 | 5 | fmpt2 7237 | . 2 ⊢ (∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2𝑜 〈𝑦, (1𝑜 ∖ 𝑧)〉 ∈ (𝐼 × 2𝑜) ↔ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) |
7 | 4, 6 | mpbi 220 | 1 ⊢ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∖ cdif 3571 〈cop 4183 × cxp 5112 ⟶wf 5884 ↦ cmpt2 6652 1𝑜c1o 7553 2𝑜c2o 7554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-1o 7560 df-2o 7561 |
This theorem is referenced by: efgtf 18135 efgtlen 18139 efginvrel2 18140 efginvrel1 18141 efgredleme 18156 efgredlemc 18158 efgcpbllemb 18168 frgp0 18173 frgpinv 18177 vrgpinv 18182 frgpnabllem1 18276 |
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