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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 5211 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | 1 | prid1 4698 | . . . . 5 ⊢ ∅ ∈ {∅, 1o} |
3 | df2o3 8117 | . . . . 5 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2912 | . . . 4 ⊢ ∅ ∈ 2o |
5 | efgval.w | . . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
6 | efgval.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
7 | 5, 6 | efgi 18845 | . . . 4 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ ∅ ∈ 2o)) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉)) |
8 | 4, 7 | mpanr2 702 | . . 3 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉)) |
9 | 8 | 3impa 1106 | . 2 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉)) |
10 | tru 1541 | . . . 4 ⊢ ⊤ | |
11 | eqidd 2822 | . . . . 5 ⊢ (⊤ → 〈𝐽, ∅〉 = 〈𝐽, ∅〉) | |
12 | dif0 4332 | . . . . . . 7 ⊢ (1o ∖ ∅) = 1o | |
13 | 12 | opeq2i 4807 | . . . . . 6 ⊢ 〈𝐽, (1o ∖ ∅)〉 = 〈𝐽, 1o〉 |
14 | 13 | a1i 11 | . . . . 5 ⊢ (⊤ → 〈𝐽, (1o ∖ ∅)〉 = 〈𝐽, 1o〉) |
15 | 11, 14 | s2eqd 14225 | . . . 4 ⊢ (⊤ → 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉 = 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉) |
16 | oteq3 4814 | . . . 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 7167 | . 2 ⊢ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1o ∖ ∅)〉”〉〉) = (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉) |
19 | 9, 18 | breqtrdi 5107 | 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 398 ∧ w3a 1083 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ∖ cdif 3933 ∅c0 4291 {cpr 4569 〈cop 4573 〈cotp 4575 class class class wbr 5066 I cid 5459 × cxp 5553 ‘cfv 6355 (class class class)co 7156 1oc1o 8095 2oc2o 8096 0cc0 10537 ...cfz 12893 ♯chash 13691 Word cword 13862 splice csplice 14111 〈“cs2 14203 ~FG cefg 18832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-substr 14003 df-pfx 14033 df-splice 14112 df-s2 14210 df-efg 18835 |
This theorem is referenced by: (None) |
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