![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > frmdval | Structured version Visualization version GIF version |
Description: Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
frmdval.m | β’ π = (freeMndβπΌ) |
frmdval.b | β’ (πΌ β π β π΅ = Word πΌ) |
frmdval.p | β’ + = ( ++ βΎ (π΅ Γ π΅)) |
Ref | Expression |
---|---|
frmdval | β’ (πΌ β π β π = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frmdval.m | . 2 β’ π = (freeMndβπΌ) | |
2 | df-frmd 18808 | . . 3 β’ freeMnd = (π β V β¦ {β¨(Baseβndx), Word πβ©, β¨(+gβndx), ( ++ βΎ (Word π Γ Word π))β©}) | |
3 | wrdeq 14526 | . . . . . 6 β’ (π = πΌ β Word π = Word πΌ) | |
4 | frmdval.b | . . . . . . 7 β’ (πΌ β π β π΅ = Word πΌ) | |
5 | 4 | eqcomd 2734 | . . . . . 6 β’ (πΌ β π β Word πΌ = π΅) |
6 | 3, 5 | sylan9eqr 2790 | . . . . 5 β’ ((πΌ β π β§ π = πΌ) β Word π = π΅) |
7 | 6 | opeq2d 4885 | . . . 4 β’ ((πΌ β π β§ π = πΌ) β β¨(Baseβndx), Word πβ© = β¨(Baseβndx), π΅β©) |
8 | 6 | sqxpeqd 5714 | . . . . . . 7 β’ ((πΌ β π β§ π = πΌ) β (Word π Γ Word π) = (π΅ Γ π΅)) |
9 | 8 | reseq2d 5989 | . . . . . 6 β’ ((πΌ β π β§ π = πΌ) β ( ++ βΎ (Word π Γ Word π)) = ( ++ βΎ (π΅ Γ π΅))) |
10 | frmdval.p | . . . . . 6 β’ + = ( ++ βΎ (π΅ Γ π΅)) | |
11 | 9, 10 | eqtr4di 2786 | . . . . 5 β’ ((πΌ β π β§ π = πΌ) β ( ++ βΎ (Word π Γ Word π)) = + ) |
12 | 11 | opeq2d 4885 | . . . 4 β’ ((πΌ β π β§ π = πΌ) β β¨(+gβndx), ( ++ βΎ (Word π Γ Word π))β© = β¨(+gβndx), + β©) |
13 | 7, 12 | preq12d 4750 | . . 3 β’ ((πΌ β π β§ π = πΌ) β {β¨(Baseβndx), Word πβ©, β¨(+gβndx), ( ++ βΎ (Word π Γ Word π))β©} = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©}) |
14 | elex 3492 | . . 3 β’ (πΌ β π β πΌ β V) | |
15 | prex 5438 | . . . 4 β’ {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©} β V | |
16 | 15 | a1i 11 | . . 3 β’ (πΌ β π β {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©} β V) |
17 | 2, 13, 14, 16 | fvmptd2 7018 | . 2 β’ (πΌ β π β (freeMndβπΌ) = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©}) |
18 | 1, 17 | eqtrid 2780 | 1 β’ (πΌ β π β π = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3473 {cpr 4634 β¨cop 4638 Γ cxp 5680 βΎ cres 5684 βcfv 6553 Word cword 14504 ++ cconcat 14560 ndxcnx 17169 Basecbs 17187 +gcplusg 17240 freeMndcfrmd 18806 |
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-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-int 4954 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-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 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-hash 14330 df-word 14505 df-frmd 18808 |
This theorem is referenced by: frmdbas 18811 frmdplusg 18813 |
Copyright terms: Public domain | W3C validator |