Theorem frmdss2 19059

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 1210	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐼 ∈ 𝑉)
2		simpl2 1211	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐽 ⊆ 𝐼)
3		sswrd 14667	. . . . . . . . 9 ⊢ (𝐽 ⊆ 𝐼 → Word 𝐽 ⊆ Word 𝐼)
4	2, 3	syl 18	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → Word 𝐽 ⊆ Word 𝐼)
5		simprr 785	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥 ∈ Word 𝐽)
6	4, 5	sseldd 3932	. . . . . . 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 19058	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word 𝐼) → (𝑀 Σg (𝑈 ∘ 𝑥)) = 𝑥)
10	1, 6, 9	syl2anc 596	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑀 Σg (𝑈 ∘ 𝑥)) = 𝑥)
11		simpl3 1212	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐴 ∈ (SubMnd‘𝑀))
12		wrdf 14663	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Word 𝐽 → 𝑥:(0..^(♯‘𝑥))⟶𝐽)
13	12	ad2antll 742	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥:(0..^(♯‘𝑥))⟶𝐽)
14	13	frnd 6718	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ran 𝑥 ⊆ 𝐽)
15		cores 6250	. . . . . . . . 9 ⊢ (ran 𝑥 ⊆ 𝐽 → ((𝑈 ↾ 𝐽) ∘ 𝑥) = (𝑈 ∘ 𝑥))
16	14, 15	syl 18	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ((𝑈 ↾ 𝐽) ∘ 𝑥) = (𝑈 ∘ 𝑥))
17	8	vrmdf 19054	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶Word 𝐼)
18	17	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝑈:𝐼⟶Word 𝐼)
19	18	ffnd 6710	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝑈 Fn 𝐼)
20		fnssres 6662	. . . . . . . . . . 11 ⊢ ((𝑈 Fn 𝐼 ∧ 𝐽 ⊆ 𝐼) → (𝑈 ↾ 𝐽) Fn 𝐽)
21	19, 2, 20	syl2an2r 698	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ↾ 𝐽) Fn 𝐽)
22		df-ima 5664	. . . . . . . . . . 11 ⊢ (𝑈 “ 𝐽) = ran (𝑈 ↾ 𝐽)
23		simprl 783	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 “ 𝐽) ⊆ 𝐴)
24	22, 23	eqsstrrid 3970	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ran (𝑈 ↾ 𝐽) ⊆ 𝐴)
25		df-f 6542	. . . . . . . . . 10 ⊢ ((𝑈 ↾ 𝐽):𝐽⟶𝐴 ↔ ((𝑈 ↾ 𝐽) Fn 𝐽 ∧ ran (𝑈 ↾ 𝐽) ⊆ 𝐴))
26	21, 24, 25	sylanbrc 595	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ↾ 𝐽):𝐽⟶𝐴)
27		wrdco 14982	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝐽 ∧ (𝑈 ↾ 𝐽):𝐽⟶𝐴) → ((𝑈 ↾ 𝐽) ∘ 𝑥) ∈ Word 𝐴)
28	5, 26, 27	syl2anc 596	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ((𝑈 ↾ 𝐽) ∘ 𝑥) ∈ Word 𝐴)
29	16, 28	eqeltrrd 2862	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ∘ 𝑥) ∈ Word 𝐴)
30		gsumwsubmcl 19033	. . . . . . 7 ⊢ ((𝐴 ∈ (SubMnd‘𝑀) ∧ (𝑈 ∘ 𝑥) ∈ Word 𝐴) → (𝑀 Σg (𝑈 ∘ 𝑥)) ∈ 𝐴)
31	11, 29, 30	syl2anc 596	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑀 Σg (𝑈 ∘ 𝑥)) ∈ 𝐴)
32	10, 31	eqeltrrd 2862	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥 ∈ 𝐴)
33	32	expr 462	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ (𝑈 “ 𝐽) ⊆ 𝐴) → (𝑥 ∈ Word 𝐽 → 𝑥 ∈ 𝐴))
34	33	ssrdv 3937	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ (𝑈 “ 𝐽) ⊆ 𝐴) → Word 𝐽 ⊆ 𝐴)
35	34	ex 418	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ 𝐴 → Word 𝐽 ⊆ 𝐴))
36		simpl1 1210	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝐼 ∈ 𝑉)
37		simp2 1155	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝐽 ⊆ 𝐼)
38	37	sselda 3931	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐼)
39	8	vrmdval 19053	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → (𝑈‘𝑥) = ⟨“𝑥”⟩)
40	36, 38, 39	syl2anc 596	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → (𝑈‘𝑥) = ⟨“𝑥”⟩)
41		simpr 490	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
42	41	s1cld 14750	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → ⟨“𝑥”⟩ ∈ Word 𝐽)
43	40, 42	eqeltrd 2861	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → (𝑈‘𝑥) ∈ Word 𝐽)
44	43	ralrimiva 3155	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ∀𝑥 ∈ 𝐽 (𝑈‘𝑥) ∈ Word 𝐽)
45	18	ffund 6714	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → Fun 𝑈)
46	18	fdmd 6720	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → dom 𝑈 = 𝐼)
47	37, 46	sseqtrrd 3968	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝐽 ⊆ dom 𝑈)
48		funimass4 6949	. . . . 5 ⊢ ((Fun 𝑈 ∧ 𝐽 ⊆ dom 𝑈) → ((𝑈 “ 𝐽) ⊆ Word 𝐽 ↔ ∀𝑥 ∈ 𝐽 (𝑈‘𝑥) ∈ Word 𝐽))
49	45, 47, 48	syl2anc 596	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ Word 𝐽 ↔ ∀𝑥 ∈ 𝐽 (𝑈‘𝑥) ∈ Word 𝐽))
50	44, 49	mpbird 260	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → (𝑈 “ 𝐽) ⊆ Word 𝐽)
51		sstr2 3938	. . 3 ⊢ ((𝑈 “ 𝐽) ⊆ Word 𝐽 → (Word 𝐽 ⊆ 𝐴 → (𝑈 “ 𝐽) ⊆ 𝐴))
52	50, 51	syl 18	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → (Word 𝐽 ⊆ 𝐴 → (𝑈 “ 𝐽) ⊆ 𝐴))
53	35, 52	impbid 215	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ 𝐴 ↔ Word 𝐽 ⊆ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  0cc0 11200  ..^cfzo 13788  ♯chash 14474  Word cword 14658  ⟨“cs1 14742   Σ_g cgsu 17611  SubMndcsubmnd 18977  freeMndcfrmd 19043  var_FMndcvrmd 19044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-n0 12607  df-xnn0 12680  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789  df-seq 14145  df-hash 14475  df-word 14659  df-lsw 14708  df-concat 14716  df-s1 14743  df-substr 14789  df-pfx 14821  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-0g 17612  df-gsum 17613  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-frmd 19045  df-vrmd 19046

This theorem is used by: (None)