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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 | ⊢ 𝑀 = (freeMnd‘𝐼) |
Ref | Expression |
---|---|
frmd0 | ⊢ ∅ = (0g‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2737 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
3 | eqid 2737 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
4 | wrd0 14094 | . . . 4 ⊢ ∅ ∈ Word 𝐼 | |
5 | frmdmnd.m | . . . . 5 ⊢ 𝑀 = (freeMnd‘𝐼) | |
6 | 5, 1 | frmdbas 18279 | . . . 4 ⊢ (𝐼 ∈ V → (Base‘𝑀) = Word 𝐼) |
7 | 4, 6 | eleqtrrid 2845 | . . 3 ⊢ (𝐼 ∈ V → ∅ ∈ (Base‘𝑀)) |
8 | 5, 1, 3 | frmdadd 18282 | . . . . 5 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥)) |
9 | 7, 8 | sylan 583 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥)) |
10 | 5, 1 | frmdelbas 18280 | . . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼) |
11 | 10 | adantl 485 | . . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼) |
12 | ccatlid 14143 | . . . . 5 ⊢ (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥) |
14 | 9, 13 | eqtrd 2777 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = 𝑥) |
15 | 5, 1, 3 | frmdadd 18282 | . . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
16 | 15 | ancoms 462 | . . . . 5 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
17 | 7, 16 | sylan 583 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
18 | ccatrid 14144 | . . . . 5 ⊢ (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥) | |
19 | 11, 18 | syl 17 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥) |
20 | 17, 19 | eqtrd 2777 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = 𝑥) |
21 | 1, 2, 3, 7, 14, 20 | ismgmid2 18140 | . 2 ⊢ (𝐼 ∈ V → ∅ = (0g‘𝑀)) |
22 | 0g0 18136 | . . 3 ⊢ ∅ = (0g‘∅) | |
23 | fvprc 6709 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (freeMnd‘𝐼) = ∅) | |
24 | 5, 23 | syl5eq 2790 | . . . 4 ⊢ (¬ 𝐼 ∈ V → 𝑀 = ∅) |
25 | 24 | fveq2d 6721 | . . 3 ⊢ (¬ 𝐼 ∈ V → (0g‘𝑀) = (0g‘∅)) |
26 | 22, 25 | eqtr4id 2797 | . 2 ⊢ (¬ 𝐼 ∈ V → ∅ = (0g‘𝑀)) |
27 | 21, 26 | pm2.61i 185 | 1 ⊢ ∅ = (0g‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∅c0 4237 ‘cfv 6380 (class class class)co 7213 Word cword 14069 ++ cconcat 14125 Basecbs 16760 +gcplusg 16802 0gc0g 16944 freeMndcfrmd 18274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-fzo 13239 df-hash 13897 df-word 14070 df-concat 14126 df-struct 16700 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-0g 16946 df-frmd 18276 |
This theorem is referenced by: frmdsssubm 18288 frmdgsum 18289 frmdup1 18291 frgpmhm 19155 mrsub0 33191 elmrsubrn 33195 |
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