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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 18782 | . . . 4 β’ ((πΌ β π β§ π β πΌ) β (πβπ) = β¨βπββ©) |
5 | 1, 2, 4 | syl2anc 583 | . . 3 β’ (π β (πβπ) = β¨βπββ©) |
6 | 5 | fveq2d 6889 | . 2 β’ (π β (πΈβ(πβπ)) = (πΈββ¨βπββ©)) |
7 | 2 | s1cld 14559 | . . . 4 β’ (π β β¨βπββ© β Word πΌ) |
8 | coeq2 5852 | . . . . . 6 β’ (π₯ = β¨βπββ© β (π΄ β π₯) = (π΄ β β¨βπββ©)) | |
9 | 8 | oveq2d 7421 | . . . . 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 7438 | . . . . 5 β’ (πΊ Ξ£g (π΄ β π₯)) β V | |
12 | 9, 10, 11 | fvmpt3i 6997 | . . . 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 14790 | . . . . 5 β’ ((π β πΌ β§ π΄:πΌβΆπ΅) β (π΄ β β¨βπββ©) = β¨β(π΄βπ)ββ©) | |
16 | 2, 14, 15 | syl2anc 583 | . . . 4 β’ (π β (π΄ β β¨βπββ©) = β¨β(π΄βπ)ββ©) |
17 | 16 | oveq2d 7421 | . . 3 β’ (π β (πΊ Ξ£g (π΄ β β¨βπββ©)) = (πΊ Ξ£g β¨β(π΄βπ)ββ©)) |
18 | 14, 2 | ffvelcdmd 7081 | . . . 4 β’ (π β (π΄βπ) β π΅) |
19 | frmdup.b | . . . . 5 β’ π΅ = (BaseβπΊ) | |
20 | 19 | gsumws1 18763 | . . . 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 2770 | . 2 β’ (π β (πΈββ¨βπββ©) = (π΄βπ)) |
23 | 6, 22 | eqtrd 2766 | 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 5224 β ccom 5673 βΆwf 6533 βcfv 6537 (class class class)co 7405 Word cword 14470 β¨βcs1 14551 Basecbs 17153 Ξ£g cgsu 17395 Mndcmnd 18667 freeMndcfrmd 18772 varFMndcvrmd 18773 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-word 14471 df-s1 14552 df-0g 17396 df-gsum 17397 df-vrmd 18775 |
This theorem is referenced by: frmdup3 18792 |
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