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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 2726 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2726 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
3 | eqid 2726 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
4 | wrd0 14547 | . . . 4 ⊢ ∅ ∈ Word 𝐼 | |
5 | frmdmnd.m | . . . . 5 ⊢ 𝑀 = (freeMnd‘𝐼) | |
6 | 5, 1 | frmdbas 18842 | . . . 4 ⊢ (𝐼 ∈ V → (Base‘𝑀) = Word 𝐼) |
7 | 4, 6 | eleqtrrid 2833 | . . 3 ⊢ (𝐼 ∈ V → ∅ ∈ (Base‘𝑀)) |
8 | 5, 1, 3 | frmdadd 18845 | . . . . 5 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥)) |
9 | 7, 8 | sylan 578 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥)) |
10 | 5, 1 | frmdelbas 18843 | . . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼) |
11 | 10 | adantl 480 | . . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼) |
12 | ccatlid 14594 | . . . . 5 ⊢ (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥) |
14 | 9, 13 | eqtrd 2766 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = 𝑥) |
15 | 5, 1, 3 | frmdadd 18845 | . . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
16 | 15 | ancoms 457 | . . . . 5 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
17 | 7, 16 | sylan 578 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
18 | ccatrid 14595 | . . . . 5 ⊢ (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥) | |
19 | 11, 18 | syl 17 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥) |
20 | 17, 19 | eqtrd 2766 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = 𝑥) |
21 | 1, 2, 3, 7, 14, 20 | ismgmid2 18661 | . 2 ⊢ (𝐼 ∈ V → ∅ = (0g‘𝑀)) |
22 | 0g0 18657 | . . 3 ⊢ ∅ = (0g‘∅) | |
23 | fvprc 6893 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (freeMnd‘𝐼) = ∅) | |
24 | 5, 23 | eqtrid 2778 | . . . 4 ⊢ (¬ 𝐼 ∈ V → 𝑀 = ∅) |
25 | 24 | fveq2d 6905 | . . 3 ⊢ (¬ 𝐼 ∈ V → (0g‘𝑀) = (0g‘∅)) |
26 | 22, 25 | eqtr4id 2785 | . 2 ⊢ (¬ 𝐼 ∈ V → ∅ = (0g‘𝑀)) |
27 | 21, 26 | pm2.61i 182 | 1 ⊢ ∅ = (0g‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 ‘cfv 6554 (class class class)co 7424 Word cword 14522 ++ cconcat 14578 Basecbs 17213 +gcplusg 17266 0gc0g 17454 freeMndcfrmd 18837 |
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 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-hash 14348 df-word 14523 df-concat 14579 df-struct 17149 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-0g 17456 df-frmd 18839 |
This theorem is referenced by: frmdsssubm 18851 frmdgsum 18852 frmdup1 18854 frgpmhm 19763 mrsub0 35344 elmrsubrn 35348 |
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