Theorem frmdss2 18823

Description: A subset of generators is contained in a submonoid iff the set of words on the generators is in the submonoid. This can be viewed as an elementary way of saying "the monoidal closure of 𝐽 is Word 𝐽". (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
frmdmnd.m	⊢ 𝑀 = (freeMnd‘𝐼)
frmdgsum.u	⊢ 𝑈 = (varFMnd‘𝐼)

Assertion

Ref	Expression
frmdss2	⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ 𝐴 ↔ Word 𝐽 ⊆ 𝐴))

Proof of Theorem frmdss2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1198	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐼 ∈ 𝑉)
2		simpl2 1199	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐽 ⊆ 𝐼)
3		sswrd 14476	. . . . . . . . 9 ⊢ (𝐽 ⊆ 𝐼 → Word 𝐽 ⊆ Word 𝐼)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → Word 𝐽 ⊆ Word 𝐼)
5		simprr 778	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥 ∈ Word 𝐽)
6	4, 5	sseldd 3916	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥 ∈ Word 𝐼)
7		frmdmnd.m	. . . . . . . 8 ⊢ 𝑀 = (freeMnd‘𝐼)
8		frmdgsum.u	. . . . . . . 8 ⊢ 𝑈 = (varFMnd‘𝐼)
9	7, 8	frmdgsum 18822	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word 𝐼) → (𝑀 Σg (𝑈 ∘ 𝑥)) = 𝑥)
10	1, 6, 9	syl2anc 590	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑀 Σg (𝑈 ∘ 𝑥)) = 𝑥)
11		simpl3 1200	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐴 ∈ (SubMnd‘𝑀))
12		wrdf 14472	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Word 𝐽 → 𝑥:(0..^(♯‘𝑥))⟶𝐽)
13	12	ad2antll 735	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥:(0..^(♯‘𝑥))⟶𝐽)
14	13	frnd 6664	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ran 𝑥 ⊆ 𝐽)
15		cores 6201	. . . . . . . . 9 ⊢ (ran 𝑥 ⊆ 𝐽 → ((𝑈 ↾ 𝐽) ∘ 𝑥) = (𝑈 ∘ 𝑥))
16	14, 15	syl 17	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ((𝑈 ↾ 𝐽) ∘ 𝑥) = (𝑈 ∘ 𝑥))
17	8	vrmdf 18818	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶Word 𝐼)
18	17	3ad2ant1 1139	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝑈:𝐼⟶Word 𝐼)
19	18	ffnd 6657	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝑈 Fn 𝐼)
20		fnssres 6609	. . . . . . . . . . 11 ⊢ ((𝑈 Fn 𝐼 ∧ 𝐽 ⊆ 𝐼) → (𝑈 ↾ 𝐽) Fn 𝐽)
21	19, 2, 20	syl2an2r 691	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ↾ 𝐽) Fn 𝐽)
22		df-ima 5632	. . . . . . . . . . 11 ⊢ (𝑈 “ 𝐽) = ran (𝑈 ↾ 𝐽)
23		simprl 776	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 “ 𝐽) ⊆ 𝐴)
24	22, 23	eqsstrrid 3954	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ran (𝑈 ↾ 𝐽) ⊆ 𝐴)
25		df-f 6490	. . . . . . . . . 10 ⊢ ((𝑈 ↾ 𝐽):𝐽⟶𝐴 ↔ ((𝑈 ↾ 𝐽) Fn 𝐽 ∧ ran (𝑈 ↾ 𝐽) ⊆ 𝐴))
26	21, 24, 25	sylanbrc 589	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ↾ 𝐽):𝐽⟶𝐴)
27		wrdco 14785	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝐽 ∧ (𝑈 ↾ 𝐽):𝐽⟶𝐴) → ((𝑈 ↾ 𝐽) ∘ 𝑥) ∈ Word 𝐴)
28	5, 26, 27	syl2anc 590	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ((𝑈 ↾ 𝐽) ∘ 𝑥) ∈ Word 𝐴)
29	16, 28	eqeltrrd 2840	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ∘ 𝑥) ∈ Word 𝐴)
30		gsumwsubmcl 18797	. . . . . . 7 ⊢ ((𝐴 ∈ (SubMnd‘𝑀) ∧ (𝑈 ∘ 𝑥) ∈ Word 𝐴) → (𝑀 Σg (𝑈 ∘ 𝑥)) ∈ 𝐴)
31	11, 29, 30	syl2anc 590	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑀 Σg (𝑈 ∘ 𝑥)) ∈ 𝐴)
32	10, 31	eqeltrrd 2840	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥 ∈ 𝐴)
33	32	expr 457	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ (𝑈 “ 𝐽) ⊆ 𝐴) → (𝑥 ∈ Word 𝐽 → 𝑥 ∈ 𝐴))
34	33	ssrdv 3921	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ (𝑈 “ 𝐽) ⊆ 𝐴) → Word 𝐽 ⊆ 𝐴)
35	34	ex 413	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ 𝐴 → Word 𝐽 ⊆ 𝐴))
36		simpl1 1198	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝐼 ∈ 𝑉)
37		simp2 1143	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝐽 ⊆ 𝐼)
38	37	sselda 3915	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐼)
39	8	vrmdval 18817	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → (𝑈‘𝑥) = ⟨“𝑥”⟩)
40	36, 38, 39	syl2anc 590	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → (𝑈‘𝑥) = ⟨“𝑥”⟩)
41		simpr 485	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
42	41	s1cld 14558	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → ⟨“𝑥”⟩ ∈ Word 𝐽)
43	40, 42	eqeltrd 2839	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → (𝑈‘𝑥) ∈ Word 𝐽)
44	43	ralrimiva 3131	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ∀𝑥 ∈ 𝐽 (𝑈‘𝑥) ∈ Word 𝐽)
45	18	ffund 6660	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → Fun 𝑈)
46	18	fdmd 6666	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → dom 𝑈 = 𝐼)
47	37, 46	sseqtrrd 3952	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝐽 ⊆ dom 𝑈)
48		funimass4 6892	. . . . 5 ⊢ ((Fun 𝑈 ∧ 𝐽 ⊆ dom 𝑈) → ((𝑈 “ 𝐽) ⊆ Word 𝐽 ↔ ∀𝑥 ∈ 𝐽 (𝑈‘𝑥) ∈ Word 𝐽))
49	45, 47, 48	syl2anc 590	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ Word 𝐽 ↔ ∀𝑥 ∈ 𝐽 (𝑈‘𝑥) ∈ Word 𝐽))
50	44, 49	mpbird 258	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → (𝑈 “ 𝐽) ⊆ Word 𝐽)
51		sstr2 3922	. . 3 ⊢ ((𝑈 “ 𝐽) ⊆ Word 𝐽 → (Word 𝐽 ⊆ 𝐴 → (𝑈 “ 𝐽) ⊆ 𝐴))
52	50, 51	syl 17	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → (Word 𝐽 ⊆ 𝐴 → (𝑈 “ 𝐽) ⊆ 𝐴))
53	35, 52	impbid 213	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ 𝐴 ↔ Word 𝐽 ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ⊆ wss 3883  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622   ∘ ccom 5623  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357  0cc0 11030  ..^cfzo 13600  ♯chash 14284  Word cword 14467  ⟨“cs1 14550   Σ_g cgsu 17395  SubMndcsubmnd 18742  freeMndcfrmd 18807  var_FMndcvrmd 18808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-card 9855  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-n0 12430  df-xnn0 12503  df-z 12517  df-uz 12781  df-fz 13454  df-fzo 13601  df-seq 13956  df-hash 14285  df-word 14468  df-lsw 14517  df-concat 14525  df-s1 14551  df-substr 14596  df-pfx 14626  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-ress 17193  df-plusg 17225  df-0g 17396  df-gsum 17397  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-submnd 18744  df-frmd 18809  df-vrmd 18810

This theorem is referenced by: (None)