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Mirrors > Home > MPE Home > Th. List > efgi1 | 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 | β’ π = ( I βWord (πΌ Γ 2o)) |
efgval.r | β’ βΌ = ( ~FG βπΌ) |
Ref | Expression |
---|---|
efgi1 | β’ ((π΄ β π β§ π β (0...(β―βπ΄)) β§ π½ β πΌ) β π΄ βΌ (π΄ splice β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, β β©ββ©β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 8426 | . . . . . 6 β’ 1o β V | |
2 | 1 | prid2 4728 | . . . . 5 β’ 1o β {β , 1o} |
3 | df2o3 8424 | . . . . 5 β’ 2o = {β , 1o} | |
4 | 2, 3 | eleqtrri 2833 | . . . 4 β’ 1o β 2o |
5 | efgval.w | . . . . 5 β’ π = ( I βWord (πΌ Γ 2o)) | |
6 | efgval.r | . . . . 5 β’ βΌ = ( ~FG βπΌ) | |
7 | 5, 6 | efgi 19509 | . . . 4 β’ (((π΄ β π β§ π β (0...(β―βπ΄))) β§ (π½ β πΌ β§ 1o β 2o)) β π΄ βΌ (π΄ splice β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, (1o β 1o)β©ββ©β©)) |
8 | 4, 7 | mpanr2 703 | . . 3 β’ (((π΄ β π β§ π β (0...(β―βπ΄))) β§ π½ β πΌ) β π΄ βΌ (π΄ splice β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, (1o β 1o)β©ββ©β©)) |
9 | 8 | 3impa 1111 | . 2 β’ ((π΄ β π β§ π β (0...(β―βπ΄)) β§ π½ β πΌ) β π΄ βΌ (π΄ splice β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, (1o β 1o)β©ββ©β©)) |
10 | tru 1546 | . . . 4 β’ β€ | |
11 | eqidd 2734 | . . . . 5 β’ (β€ β β¨π½, 1oβ© = β¨π½, 1oβ©) | |
12 | difid 4334 | . . . . . . 7 β’ (1o β 1o) = β | |
13 | 12 | opeq2i 4838 | . . . . . 6 β’ β¨π½, (1o β 1o)β© = β¨π½, β β© |
14 | 13 | a1i 11 | . . . . 5 β’ (β€ β β¨π½, (1o β 1o)β© = β¨π½, β β©) |
15 | 11, 14 | s2eqd 14761 | . . . 4 β’ (β€ β β¨ββ¨π½, 1oβ©β¨π½, (1o β 1o)β©ββ© = β¨ββ¨π½, 1oβ©β¨π½, β β©ββ©) |
16 | oteq3 4845 | . . . 4 β’ (β¨ββ¨π½, 1oβ©β¨π½, (1o β 1o)β©ββ© = β¨ββ¨π½, 1oβ©β¨π½, β β©ββ© β β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, (1o β 1o)β©ββ©β© = β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, β β©ββ©β©) | |
17 | 10, 15, 16 | mp2b 10 | . . 3 β’ β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, (1o β 1o)β©ββ©β© = β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, β β©ββ©β© |
18 | 17 | oveq2i 7372 | . 2 β’ (π΄ splice β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, (1o β 1o)β©ββ©β©) = (π΄ splice β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, β β©ββ©β©) |
19 | 9, 18 | breqtrdi 5150 | 1 β’ ((π΄ β π β§ π β (0...(β―βπ΄)) β§ π½ β πΌ) β π΄ βΌ (π΄ splice β¨π, π, β¨ββ¨π½, 1oβ©β¨π½, β β©ββ©β©)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β€wtru 1543 β wcel 2107 β cdif 3911 β c0 4286 {cpr 4592 β¨cop 4596 β¨cotp 4598 class class class wbr 5109 I cid 5534 Γ cxp 5635 βcfv 6500 (class class class)co 7361 1oc1o 8409 2oc2o 8410 0cc0 11059 ...cfz 13433 β―chash 14239 Word cword 14411 splice csplice 14646 β¨βcs2 14739 ~FG cefg 19496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-ot 4599 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-concat 14468 df-s1 14493 df-substr 14538 df-pfx 14568 df-splice 14647 df-s2 14746 df-efg 19499 |
This theorem is referenced by: (None) |
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