Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | ⊢ 𝑀 = (freeMnd‘𝐼) |
Ref | Expression |
---|---|
frmd0 | ⊢ ∅ = (0g‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2622 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
3 | eqid 2622 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
4 | wrd0 13330 | . . . 4 ⊢ ∅ ∈ Word 𝐼 | |
5 | frmdmnd.m | . . . . 5 ⊢ 𝑀 = (freeMnd‘𝐼) | |
6 | 5, 1 | frmdbas 17389 | . . . 4 ⊢ (𝐼 ∈ V → (Base‘𝑀) = Word 𝐼) |
7 | 4, 6 | syl5eleqr 2708 | . . 3 ⊢ (𝐼 ∈ V → ∅ ∈ (Base‘𝑀)) |
8 | 5, 1, 3 | frmdadd 17392 | . . . . 5 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥)) |
9 | 7, 8 | sylan 488 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥)) |
10 | 5, 1 | frmdelbas 17390 | . . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼) |
11 | 10 | adantl 482 | . . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼) |
12 | ccatlid 13369 | . . . . 5 ⊢ (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥) |
14 | 9, 13 | eqtrd 2656 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = 𝑥) |
15 | 5, 1, 3 | frmdadd 17392 | . . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
16 | 15 | ancoms 469 | . . . . 5 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
17 | 7, 16 | sylan 488 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
18 | ccatrid 13370 | . . . . 5 ⊢ (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥) | |
19 | 11, 18 | syl 17 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥) |
20 | 17, 19 | eqtrd 2656 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = 𝑥) |
21 | 1, 2, 3, 7, 14, 20 | ismgmid2 17267 | . 2 ⊢ (𝐼 ∈ V → ∅ = (0g‘𝑀)) |
22 | fvprc 6185 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (freeMnd‘𝐼) = ∅) | |
23 | 5, 22 | syl5eq 2668 | . . . 4 ⊢ (¬ 𝐼 ∈ V → 𝑀 = ∅) |
24 | 23 | fveq2d 6195 | . . 3 ⊢ (¬ 𝐼 ∈ V → (0g‘𝑀) = (0g‘∅)) |
25 | 0g0 17263 | . . 3 ⊢ ∅ = (0g‘∅) | |
26 | 24, 25 | syl6reqr 2675 | . 2 ⊢ (¬ 𝐼 ∈ V → ∅ = (0g‘𝑀)) |
27 | 21, 26 | pm2.61i 176 | 1 ⊢ ∅ = (0g‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 Word cword 13291 ++ cconcat 13293 Basecbs 15857 +gcplusg 15941 0gc0g 16100 freeMndcfrmd 17384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-0g 16102 df-frmd 17386 |
This theorem is referenced by: frmdsssubm 17398 frmdgsum 17399 frmdup1 17401 frgpmhm 18178 mrsub0 31413 elmrsubrn 31417 |
Copyright terms: Public domain | W3C validator |