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Mirrors > Home > MPE Home > Th. List > frmd0 | Structured version Visualization version GIF version |
Description: The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
frmdmnd.m | β’ π = (freeMndβπΌ) |
Ref | Expression |
---|---|
frmd0 | β’ β = (0gβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2728 | . . 3 β’ (0gβπ) = (0gβπ) | |
3 | eqid 2728 | . . 3 β’ (+gβπ) = (+gβπ) | |
4 | wrd0 14522 | . . . 4 β’ β β Word πΌ | |
5 | frmdmnd.m | . . . . 5 β’ π = (freeMndβπΌ) | |
6 | 5, 1 | frmdbas 18804 | . . . 4 β’ (πΌ β V β (Baseβπ) = Word πΌ) |
7 | 4, 6 | eleqtrrid 2836 | . . 3 β’ (πΌ β V β β β (Baseβπ)) |
8 | 5, 1, 3 | frmdadd 18807 | . . . . 5 β’ ((β β (Baseβπ) β§ π₯ β (Baseβπ)) β (β (+gβπ)π₯) = (β ++ π₯)) |
9 | 7, 8 | sylan 579 | . . . 4 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (β (+gβπ)π₯) = (β ++ π₯)) |
10 | 5, 1 | frmdelbas 18805 | . . . . . 6 β’ (π₯ β (Baseβπ) β π₯ β Word πΌ) |
11 | 10 | adantl 481 | . . . . 5 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β π₯ β Word πΌ) |
12 | ccatlid 14569 | . . . . 5 β’ (π₯ β Word πΌ β (β ++ π₯) = π₯) | |
13 | 11, 12 | syl 17 | . . . 4 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (β ++ π₯) = π₯) |
14 | 9, 13 | eqtrd 2768 | . . 3 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (β (+gβπ)π₯) = π₯) |
15 | 5, 1, 3 | frmdadd 18807 | . . . . . 6 β’ ((π₯ β (Baseβπ) β§ β β (Baseβπ)) β (π₯(+gβπ)β ) = (π₯ ++ β )) |
16 | 15 | ancoms 458 | . . . . 5 β’ ((β β (Baseβπ) β§ π₯ β (Baseβπ)) β (π₯(+gβπ)β ) = (π₯ ++ β )) |
17 | 7, 16 | sylan 579 | . . . 4 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (π₯(+gβπ)β ) = (π₯ ++ β )) |
18 | ccatrid 14570 | . . . . 5 β’ (π₯ β Word πΌ β (π₯ ++ β ) = π₯) | |
19 | 11, 18 | syl 17 | . . . 4 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (π₯ ++ β ) = π₯) |
20 | 17, 19 | eqtrd 2768 | . . 3 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (π₯(+gβπ)β ) = π₯) |
21 | 1, 2, 3, 7, 14, 20 | ismgmid2 18628 | . 2 β’ (πΌ β V β β = (0gβπ)) |
22 | 0g0 18624 | . . 3 β’ β = (0gββ ) | |
23 | fvprc 6889 | . . . . 5 β’ (Β¬ πΌ β V β (freeMndβπΌ) = β ) | |
24 | 5, 23 | eqtrid 2780 | . . . 4 β’ (Β¬ πΌ β V β π = β ) |
25 | 24 | fveq2d 6901 | . . 3 β’ (Β¬ πΌ β V β (0gβπ) = (0gββ )) |
26 | 22, 25 | eqtr4id 2787 | . 2 β’ (Β¬ πΌ β V β β = (0gβπ)) |
27 | 21, 26 | pm2.61i 182 | 1 β’ β = (0gβπ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3471 β c0 4323 βcfv 6548 (class class class)co 7420 Word cword 14497 ++ cconcat 14553 Basecbs 17180 +gcplusg 17233 0gc0g 17421 freeMndcfrmd 18799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-concat 14554 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-0g 17423 df-frmd 18801 |
This theorem is referenced by: frmdsssubm 18813 frmdgsum 18814 frmdup1 18816 frgpmhm 19720 mrsub0 35126 elmrsubrn 35130 |
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