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Theorem frmdmnd 18669
Description: A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypothesis
Ref Expression
frmdmnd.m 𝑀 = (freeMnd‘𝐼)
Assertion
Ref Expression
frmdmnd (𝐼𝑉𝑀 ∈ Mnd)

Proof of Theorem frmdmnd
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2737 . 2 (𝐼𝑉 → (Base‘𝑀) = (Base‘𝑀))
2 eqidd 2737 . 2 (𝐼𝑉 → (+g𝑀) = (+g𝑀))
3 frmdmnd.m . . . . . 6 𝑀 = (freeMnd‘𝐼)
4 eqid 2736 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
5 eqid 2736 . . . . . 6 (+g𝑀) = (+g𝑀)
63, 4, 5frmdadd 18665 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) = (𝑥 ++ 𝑦))
73, 4frmdelbas 18663 . . . . . 6 (𝑥 ∈ (Base‘𝑀) → 𝑥 ∈ Word 𝐼)
83, 4frmdelbas 18663 . . . . . 6 (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
9 ccatcl 14462 . . . . . 6 ((𝑥 ∈ Word 𝐼𝑦 ∈ Word 𝐼) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
107, 8, 9syl2an 596 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
116, 10eqeltrd 2838 . . . 4 ((𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ Word 𝐼)
12113adant1 1130 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ Word 𝐼)
133, 4frmdbas 18662 . . . 4 (𝐼𝑉 → (Base‘𝑀) = Word 𝐼)
14133ad2ant1 1133 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (Base‘𝑀) = Word 𝐼)
1512, 14eleqtrrd 2841 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
16 simpr1 1194 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
1716, 7syl 17 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑥 ∈ Word 𝐼)
18 simpr2 1195 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
1918, 8syl 17 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
20 simpr3 1196 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ (Base‘𝑀))
213, 4frmdelbas 18663 . . . . . 6 (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
2220, 21syl 17 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
23 ccatass 14476 . . . . 5 ((𝑥 ∈ Word 𝐼𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
2417, 19, 22, 23syl3anc 1371 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦) ++ 𝑧) = (𝑥 ++ (𝑦 ++ 𝑧)))
2516, 18, 10syl2anc 584 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ Word 𝐼)
2613adantr 481 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (Base‘𝑀) = Word 𝐼)
2725, 26eleqtrrd 2841 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥 ++ 𝑦) ∈ (Base‘𝑀))
283, 4, 5frmdadd 18665 . . . . 5 (((𝑥 ++ 𝑦) ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
2927, 20, 28syl2anc 584 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦) ++ 𝑧))
30 ccatcl 14462 . . . . . . 7 ((𝑦 ∈ Word 𝐼𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
3119, 22, 30syl2anc 584 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
3231, 26eleqtrrd 2841 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ (Base‘𝑀))
333, 4, 5frmdadd 18665 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ (𝑦 ++ 𝑧) ∈ (Base‘𝑀)) → (𝑥(+g𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
3416, 32, 33syl2anc 584 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)(𝑦 ++ 𝑧)) = (𝑥 ++ (𝑦 ++ 𝑧)))
3524, 29, 343eqtr4d 2786 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥 ++ 𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦 ++ 𝑧)))
3616, 18, 6syl2anc 584 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) = (𝑥 ++ 𝑦))
3736oveq1d 7372 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = ((𝑥 ++ 𝑦)(+g𝑀)𝑧))
383, 4, 5frmdadd 18665 . . . . 5 ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
3918, 20, 38syl2anc 584 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+g𝑀)𝑧) = (𝑦 ++ 𝑧))
4039oveq2d 7373 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)) = (𝑥(+g𝑀)(𝑦 ++ 𝑧)))
4135, 37, 403eqtr4d 2786 . 2 ((𝐼𝑉 ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝑥(+g𝑀)𝑦)(+g𝑀)𝑧) = (𝑥(+g𝑀)(𝑦(+g𝑀)𝑧)))
42 wrd0 14427 . . 3 ∅ ∈ Word 𝐼
4342, 13eleqtrrid 2845 . 2 (𝐼𝑉 → ∅ ∈ (Base‘𝑀))
443, 4, 5frmdadd 18665 . . . 4 ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = (∅ ++ 𝑥))
4543, 44sylan 580 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = (∅ ++ 𝑥))
467adantl 482 . . . 4 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → 𝑥 ∈ Word 𝐼)
47 ccatlid 14474 . . . 4 (𝑥 ∈ Word 𝐼 → (∅ ++ 𝑥) = 𝑥)
4846, 47syl 17 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅ ++ 𝑥) = 𝑥)
4945, 48eqtrd 2776 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (∅(+g𝑀)𝑥) = 𝑥)
503, 4, 5frmdadd 18665 . . . . 5 ((𝑥 ∈ (Base‘𝑀) ∧ ∅ ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
5150ancoms 459 . . . 4 ((∅ ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
5243, 51sylan 580 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = (𝑥 ++ ∅))
53 ccatrid 14475 . . . 4 (𝑥 ∈ Word 𝐼 → (𝑥 ++ ∅) = 𝑥)
5446, 53syl 17 . . 3 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥 ++ ∅) = 𝑥)
5552, 54eqtrd 2776 . 2 ((𝐼𝑉𝑥 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)∅) = 𝑥)
561, 2, 15, 41, 43, 49, 55ismndd 18578 1 (𝐼𝑉𝑀 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  c0 4282  cfv 6496  (class class class)co 7357  Word cword 14402   ++ cconcat 14458  Basecbs 17083  +gcplusg 17133  Mndcmnd 18556  freeMndcfrmd 18657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-frmd 18659
This theorem is referenced by:  frmdsssubm  18671  frmdgsum  18672  frmdup1  18674  frgp0  19542  frgpadd  19545  frgpmhm  19547  mrsubff  34106  mrsubccat  34112  elmrsubrn  34114
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