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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 2823 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
2 | eqid 2823 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
3 | eqid 2823 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
4 | wrd0 13891 | . . . 4 ⊢ ∅ ∈ Word 𝐼 | |
5 | frmdmnd.m | . . . . 5 ⊢ 𝑀 = (freeMnd‘𝐼) | |
6 | 5, 1 | frmdbas 18019 | . . . 4 ⊢ (𝐼 ∈ V → (Base‘𝑀) = Word 𝐼) |
7 | 4, 6 | eleqtrrid 2922 | . . 3 ⊢ (𝐼 ∈ V → ∅ ∈ (Base‘𝑀)) |
8 | 5, 1, 3 | frmdadd 18022 | . . . . 5 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥)) |
9 | 7, 8 | sylan 582 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = (∅ ++ 𝑥)) |
10 | 5, 1 | frmdelbas 18020 | . . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼) |
11 | 10 | adantl 484 | . . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼) |
12 | ccatlid 13942 | . . . . 5 ⊢ (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥) |
14 | 9, 13 | eqtrd 2858 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g‘𝑀)𝑥) = 𝑥) |
15 | 5, 1, 3 | frmdadd 18022 | . . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
16 | 15 | ancoms 461 | . . . . 5 ⊢ ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
17 | 7, 16 | sylan 582 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = (𝑥 ++ ∅)) |
18 | ccatrid 13943 | . . . . 5 ⊢ (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥) | |
19 | 11, 18 | syl 17 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥) |
20 | 17, 19 | eqtrd 2858 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)∅) = 𝑥) |
21 | 1, 2, 3, 7, 14, 20 | ismgmid2 17880 | . 2 ⊢ (𝐼 ∈ V → ∅ = (0g‘𝑀)) |
22 | fvprc 6665 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (freeMnd‘𝐼) = ∅) | |
23 | 5, 22 | syl5eq 2870 | . . . 4 ⊢ (¬ 𝐼 ∈ V → 𝑀 = ∅) |
24 | 23 | fveq2d 6676 | . . 3 ⊢ (¬ 𝐼 ∈ V → (0g‘𝑀) = (0g‘∅)) |
25 | 0g0 17876 | . . 3 ⊢ ∅ = (0g‘∅) | |
26 | 24, 25 | syl6reqr 2877 | . 2 ⊢ (¬ 𝐼 ∈ V → ∅ = (0g‘𝑀)) |
27 | 21, 26 | pm2.61i 184 | 1 ⊢ ∅ = (0g‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 ‘cfv 6357 (class class class)co 7158 Word cword 13864 ++ cconcat 13924 Basecbs 16485 +gcplusg 16567 0gc0g 16715 freeMndcfrmd 18014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-0g 16717 df-frmd 18016 |
This theorem is referenced by: frmdsssubm 18028 frmdgsum 18029 frmdup1 18031 frgpmhm 18893 mrsub0 32765 elmrsubrn 32769 |
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