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Mirrors > Home > MPE Home > Th. List > frmdup2 | Structured version Visualization version GIF version |
Description: The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
frmdup.m | β’ π = (freeMndβπΌ) |
frmdup.b | β’ π΅ = (BaseβπΊ) |
frmdup.e | β’ πΈ = (π₯ β Word πΌ β¦ (πΊ Ξ£g (π΄ β π₯))) |
frmdup.g | β’ (π β πΊ β Mnd) |
frmdup.i | β’ (π β πΌ β π) |
frmdup.a | β’ (π β π΄:πΌβΆπ΅) |
frmdup2.u | β’ π = (varFMndβπΌ) |
frmdup2.y | β’ (π β π β πΌ) |
Ref | Expression |
---|---|
frmdup2 | β’ (π β (πΈβ(πβπ)) = (π΄βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frmdup.i | . . . 4 β’ (π β πΌ β π) | |
2 | frmdup2.y | . . . 4 β’ (π β π β πΌ) | |
3 | frmdup2.u | . . . . 5 β’ π = (varFMndβπΌ) | |
4 | 3 | vrmdval 18816 | . . . 4 β’ ((πΌ β π β§ π β πΌ) β (πβπ) = β¨βπββ©) |
5 | 1, 2, 4 | syl2anc 582 | . . 3 β’ (π β (πβπ) = β¨βπββ©) |
6 | 5 | fveq2d 6906 | . 2 β’ (π β (πΈβ(πβπ)) = (πΈββ¨βπββ©)) |
7 | 2 | s1cld 14593 | . . . 4 β’ (π β β¨βπββ© β Word πΌ) |
8 | coeq2 5865 | . . . . . 6 β’ (π₯ = β¨βπββ© β (π΄ β π₯) = (π΄ β β¨βπββ©)) | |
9 | 8 | oveq2d 7442 | . . . . 5 β’ (π₯ = β¨βπββ© β (πΊ Ξ£g (π΄ β π₯)) = (πΊ Ξ£g (π΄ β β¨βπββ©))) |
10 | frmdup.e | . . . . 5 β’ πΈ = (π₯ β Word πΌ β¦ (πΊ Ξ£g (π΄ β π₯))) | |
11 | ovex 7459 | . . . . 5 β’ (πΊ Ξ£g (π΄ β π₯)) β V | |
12 | 9, 10, 11 | fvmpt3i 7015 | . . . 4 β’ (β¨βπββ© β Word πΌ β (πΈββ¨βπββ©) = (πΊ Ξ£g (π΄ β β¨βπββ©))) |
13 | 7, 12 | syl 17 | . . 3 β’ (π β (πΈββ¨βπββ©) = (πΊ Ξ£g (π΄ β β¨βπββ©))) |
14 | frmdup.a | . . . . 5 β’ (π β π΄:πΌβΆπ΅) | |
15 | s1co 14824 | . . . . 5 β’ ((π β πΌ β§ π΄:πΌβΆπ΅) β (π΄ β β¨βπββ©) = β¨β(π΄βπ)ββ©) | |
16 | 2, 14, 15 | syl2anc 582 | . . . 4 β’ (π β (π΄ β β¨βπββ©) = β¨β(π΄βπ)ββ©) |
17 | 16 | oveq2d 7442 | . . 3 β’ (π β (πΊ Ξ£g (π΄ β β¨βπββ©)) = (πΊ Ξ£g β¨β(π΄βπ)ββ©)) |
18 | 14, 2 | ffvelcdmd 7100 | . . . 4 β’ (π β (π΄βπ) β π΅) |
19 | frmdup.b | . . . . 5 β’ π΅ = (BaseβπΊ) | |
20 | 19 | gsumws1 18797 | . . . 4 β’ ((π΄βπ) β π΅ β (πΊ Ξ£g β¨β(π΄βπ)ββ©) = (π΄βπ)) |
21 | 18, 20 | syl 17 | . . 3 β’ (π β (πΊ Ξ£g β¨β(π΄βπ)ββ©) = (π΄βπ)) |
22 | 13, 17, 21 | 3eqtrd 2772 | . 2 β’ (π β (πΈββ¨βπββ©) = (π΄βπ)) |
23 | 6, 22 | eqtrd 2768 | 1 β’ (π β (πΈβ(πβπ)) = (π΄βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β¦ cmpt 5235 β ccom 5686 βΆwf 6549 βcfv 6553 (class class class)co 7426 Word cword 14504 β¨βcs1 14585 Basecbs 17187 Ξ£g cgsu 17429 Mndcmnd 18701 freeMndcfrmd 18806 varFMndcvrmd 18807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-word 14505 df-s1 14586 df-0g 17430 df-gsum 17431 df-vrmd 18809 |
This theorem is referenced by: frmdup3 18826 |
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