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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 4984 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | 1 | prid1 4486 | . . . . 5 ⊢ ∅ ∈ {∅, 1𝑜} |
3 | df2o3 7813 | . . . . 5 ⊢ 2𝑜 = {∅, 1𝑜} | |
4 | 2, 3 | eleqtrri 2877 | . . . 4 ⊢ ∅ ∈ 2𝑜 |
5 | efgval.w | . . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜)) | |
6 | efgval.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
7 | 5, 6 | efgi 18445 | . . . 4 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ ∅ ∈ 2𝑜)) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1𝑜 ∖ ∅)〉”〉〉)) |
8 | 4, 7 | mpanr2 696 | . . 3 ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1𝑜 ∖ ∅)〉”〉〉)) |
9 | 8 | 3impa 1137 | . 2 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1𝑜 ∖ ∅)〉”〉〉)) |
10 | tru 1658 | . . . 4 ⊢ ⊤ | |
11 | eqidd 2800 | . . . . 5 ⊢ (⊤ → 〈𝐽, ∅〉 = 〈𝐽, ∅〉) | |
12 | dif0 4151 | . . . . . . 7 ⊢ (1𝑜 ∖ ∅) = 1𝑜 | |
13 | 12 | opeq2i 4597 | . . . . . 6 ⊢ 〈𝐽, (1𝑜 ∖ ∅)〉 = 〈𝐽, 1𝑜〉 |
14 | 13 | a1i 11 | . . . . 5 ⊢ (⊤ → 〈𝐽, (1𝑜 ∖ ∅)〉 = 〈𝐽, 1𝑜〉) |
15 | 11, 14 | s2eqd 13948 | . . . 4 ⊢ (⊤ → 〈“〈𝐽, ∅〉〈𝐽, (1𝑜 ∖ ∅)〉”〉 = 〈“〈𝐽, ∅〉〈𝐽, 1𝑜〉”〉) |
16 | oteq3 4604 | . . . 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 6889 | . 2 ⊢ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, (1𝑜 ∖ ∅)〉”〉〉) = (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1𝑜〉”〉〉) |
19 | 9, 18 | syl6breq 4884 | 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 385 ∧ w3a 1108 = wceq 1653 ⊤wtru 1654 ∈ wcel 2157 ∖ cdif 3766 ∅c0 4115 {cpr 4370 〈cop 4374 〈cotp 4376 class class class wbr 4843 I cid 5219 × cxp 5310 ‘cfv 6101 (class class class)co 6878 1𝑜c1o 7792 2𝑜c2o 7793 0cc0 10224 ...cfz 12580 ♯chash 13370 Word cword 13534 splice csplice 13819 〈“cs2 13926 ~FG cefg 18432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-ot 4377 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-hash 13371 df-word 13535 df-concat 13591 df-s1 13616 df-substr 13665 df-pfx 13714 df-splice 13821 df-s2 13933 df-efg 18435 |
This theorem is referenced by: (None) |
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