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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 2737 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2737 | . . 3 β’ (0gβπ) = (0gβπ) | |
3 | eqid 2737 | . . 3 β’ (+gβπ) = (+gβπ) | |
4 | wrd0 14428 | . . . 4 β’ β β Word πΌ | |
5 | frmdmnd.m | . . . . 5 β’ π = (freeMndβπΌ) | |
6 | 5, 1 | frmdbas 18663 | . . . 4 β’ (πΌ β V β (Baseβπ) = Word πΌ) |
7 | 4, 6 | eleqtrrid 2845 | . . 3 β’ (πΌ β V β β β (Baseβπ)) |
8 | 5, 1, 3 | frmdadd 18666 | . . . . 5 β’ ((β β (Baseβπ) β§ π₯ β (Baseβπ)) β (β (+gβπ)π₯) = (β ++ π₯)) |
9 | 7, 8 | sylan 581 | . . . 4 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (β (+gβπ)π₯) = (β ++ π₯)) |
10 | 5, 1 | frmdelbas 18664 | . . . . . 6 β’ (π₯ β (Baseβπ) β π₯ β Word πΌ) |
11 | 10 | adantl 483 | . . . . 5 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β π₯ β Word πΌ) |
12 | ccatlid 14475 | . . . . 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 2777 | . . 3 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (β (+gβπ)π₯) = π₯) |
15 | 5, 1, 3 | frmdadd 18666 | . . . . . 6 β’ ((π₯ β (Baseβπ) β§ β β (Baseβπ)) β (π₯(+gβπ)β ) = (π₯ ++ β )) |
16 | 15 | ancoms 460 | . . . . 5 β’ ((β β (Baseβπ) β§ π₯ β (Baseβπ)) β (π₯(+gβπ)β ) = (π₯ ++ β )) |
17 | 7, 16 | sylan 581 | . . . 4 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (π₯(+gβπ)β ) = (π₯ ++ β )) |
18 | ccatrid 14476 | . . . . 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 2777 | . . 3 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (π₯(+gβπ)β ) = π₯) |
21 | 1, 2, 3, 7, 14, 20 | ismgmid2 18524 | . 2 β’ (πΌ β V β β = (0gβπ)) |
22 | 0g0 18520 | . . 3 β’ β = (0gββ ) | |
23 | fvprc 6835 | . . . . 5 β’ (Β¬ πΌ β V β (freeMndβπΌ) = β ) | |
24 | 5, 23 | eqtrid 2789 | . . . 4 β’ (Β¬ πΌ β V β π = β ) |
25 | 24 | fveq2d 6847 | . . 3 β’ (Β¬ πΌ β V β (0gβπ) = (0gββ )) |
26 | 22, 25 | eqtr4id 2796 | . 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 397 = wceq 1542 β wcel 2107 Vcvv 3446 β c0 4283 βcfv 6497 (class class class)co 7358 Word cword 14403 ++ cconcat 14459 Basecbs 17084 +gcplusg 17134 0gc0g 17322 freeMndcfrmd 18658 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-fzo 13569 df-hash 14232 df-word 14404 df-concat 14460 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-0g 17324 df-frmd 18660 |
This theorem is referenced by: frmdsssubm 18672 frmdgsum 18673 frmdup1 18675 frgpmhm 19548 mrsub0 34113 elmrsubrn 34117 |
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