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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 2724 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2724 | . . 3 β’ (0gβπ) = (0gβπ) | |
3 | eqid 2724 | . . 3 β’ (+gβπ) = (+gβπ) | |
4 | wrd0 14491 | . . . 4 β’ β β Word πΌ | |
5 | frmdmnd.m | . . . . 5 β’ π = (freeMndβπΌ) | |
6 | 5, 1 | frmdbas 18773 | . . . 4 β’ (πΌ β V β (Baseβπ) = Word πΌ) |
7 | 4, 6 | eleqtrrid 2832 | . . 3 β’ (πΌ β V β β β (Baseβπ)) |
8 | 5, 1, 3 | frmdadd 18776 | . . . . 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 18774 | . . . . . 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 14538 | . . . . 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 2764 | . . 3 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (β (+gβπ)π₯) = π₯) |
15 | 5, 1, 3 | frmdadd 18776 | . . . . . 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 14539 | . . . . 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 2764 | . . 3 β’ ((πΌ β V β§ π₯ β (Baseβπ)) β (π₯(+gβπ)β ) = π₯) |
21 | 1, 2, 3, 7, 14, 20 | ismgmid2 18597 | . 2 β’ (πΌ β V β β = (0gβπ)) |
22 | 0g0 18593 | . . 3 β’ β = (0gββ ) | |
23 | fvprc 6874 | . . . . 5 β’ (Β¬ πΌ β V β (freeMndβπΌ) = β ) | |
24 | 5, 23 | eqtrid 2776 | . . . 4 β’ (Β¬ πΌ β V β π = β ) |
25 | 24 | fveq2d 6886 | . . 3 β’ (Β¬ πΌ β V β (0gβπ) = (0gββ )) |
26 | 22, 25 | eqtr4id 2783 | . 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 1533 β wcel 2098 Vcvv 3466 β c0 4315 βcfv 6534 (class class class)co 7402 Word cword 14466 ++ cconcat 14522 Basecbs 17149 +gcplusg 17202 0gc0g 17390 freeMndcfrmd 18768 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-fzo 13629 df-hash 14292 df-word 14467 df-concat 14523 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-frmd 18770 |
This theorem is referenced by: frmdsssubm 18782 frmdgsum 18783 frmdup1 18785 frgpmhm 19681 mrsub0 35025 elmrsubrn 35029 |
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