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Mirrors > Home > MPE Home > Th. List > efgi0 | Structured version Visualization version GIF version |
Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
efgval.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) |
efgval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
Ref | Expression |
---|---|
efgi0 | ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5175 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | 1 | prid1 4658 | . . . . 5 ⊢ ∅ ∈ {∅, 1o} |
3 | df2o3 8100 | . . . . 5 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2889 | . . . 4 ⊢ ∅ ∈ 2o |
5 | efgval.w | . . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
6 | efgval.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
7 | 5, 6 | efgi 18837 | . . . 4 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ ∅ ∈ 2o)) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉)) |
8 | 4, 7 | mpanr2 703 | . . 3 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉)) |
9 | 8 | 3impa 1107 | . 2 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉)) |
10 | tru 1542 | . . . 4 ⊢ ⊤ | |
11 | eqidd 2799 | . . . . 5 ⊢ (⊤ → 〈𝐽, ∅〉 = 〈𝐽, ∅〉) | |
12 | dif0 4286 | . . . . . . 7 ⊢ (1o ∖ ∅) = 1o | |
13 | 12 | opeq2i 4769 | . . . . . 6 ⊢ 〈𝐽, (1o ∖ ∅)〉 = 〈𝐽, 1o〉 |
14 | 13 | a1i 11 | . . . . 5 ⊢ (⊤ → 〈𝐽, (1o ∖ ∅)〉 = 〈𝐽, 1o〉) |
15 | 11, 14 | s2eqd 14216 | . . . 4 ⊢ (⊤ → 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉 = 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉) |
16 | oteq3 4776 | . . . 4 ⊢ (〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉 = 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉 → 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉 = 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉) | |
17 | 10, 15, 16 | mp2b 10 | . . 3 ⊢ 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉 = 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉 |
18 | 17 | oveq2i 7146 | . 2 ⊢ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉) = (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉) |
19 | 9, 18 | breqtrdi 5071 | 1 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ∖ cdif 3878 ∅c0 4243 {cpr 4527 〈cop 4531 〈cotp 4533 class class class wbr 5030 I cid 5424 × cxp 5517 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 2oc2o 8079 0cc0 10526 ...cfz 12885 ♯chash 13686 Word cword 13857 splice csplice 14102 〈“cs2 14194 ~FG cefg 18824 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-splice 14103 df-s2 14201 df-efg 18827 |
This theorem is referenced by: (None) |
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