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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 2610 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2610 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
3 | eqid 2610 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
4 | wrd0 13185 | . . . 4 ⊢ ∅ ∈ Word 𝐼 | |
5 | frmdmnd.m | . . . . 5 ⊢ 𝑀 = (freeMnd‘𝐼) | |
6 | 5, 1 | frmdbas 17212 | . . . 4 ⊢ (𝐼 ∈ V → (Base‘𝑀) = Word 𝐼) |
7 | 4, 6 | syl5eleqr 2695 | . . 3 ⊢ (𝐼 ∈ V → ∅ ∈ (Base‘𝑀)) |
8 | 5, 1, 3 | frmdadd 17215 | . . . . 5 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥)) |
9 | 7, 8 | sylan 487 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥)) |
10 | 5, 1 | frmdelbas 17213 | . . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼) |
11 | 10 | adantl 481 | . . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼) |
12 | ccatlid 13222 | . . . . 5 ⊢ (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥) |
14 | 9, 13 | eqtrd 2644 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = 𝑥) |
15 | 5, 1, 3 | frmdadd 17215 | . . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
16 | 15 | ancoms 468 | . . . . 5 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
17 | 7, 16 | sylan 487 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
18 | ccatrid 13223 | . . . . 5 ⊢ (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥) | |
19 | 11, 18 | syl 17 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥) |
20 | 17, 19 | eqtrd 2644 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = 𝑥) |
21 | 1, 2, 3, 7, 14, 20 | ismgmid2 17090 | . 2 ⊢ (𝐼 ∈ V → ∅ = (0g‘𝑀)) |
22 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (freeMnd‘𝐼) = ∅) | |
23 | 5, 22 | syl5eq 2656 | . . . 4 ⊢ (¬ 𝐼 ∈ V → 𝑀 = ∅) |
24 | 23 | fveq2d 6107 | . . 3 ⊢ (¬ 𝐼 ∈ V → (0g‘𝑀) = (0g‘∅)) |
25 | 0g0 17086 | . . 3 ⊢ ∅ = (0g‘∅) | |
26 | 24, 25 | syl6reqr 2663 | . 2 ⊢ (¬ 𝐼 ∈ V → ∅ = (0g‘𝑀)) |
27 | 21, 26 | pm2.61i 175 | 1 ⊢ ∅ = (0g‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Word cword 13146 ++ cconcat 13148 Basecbs 15695 +gcplusg 15768 0gc0g 15923 freeMndcfrmd 17207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-0g 15925 df-frmd 17209 |
This theorem is referenced by: frmdsssubm 17221 frmdgsum 17222 frmdup1 17224 frgpmhm 18001 mrsub0 30667 elmrsubrn 30671 |
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