Theorem frmdss2 18772

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 1192	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐼 ∈ 𝑉)
2		simpl2 1193	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐽 ⊆ 𝐼)
3		sswrd 14463	. . . . . . . . 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 772	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥 ∈ Word 𝐽)
6	4, 5	sseldd 3944	. . . . . . 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 18771	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ Word 𝐼) → (𝑀 Σg (𝑈 ∘ 𝑥)) = 𝑥)
10	1, 6, 9	syl2anc 584	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑀 Σg (𝑈 ∘ 𝑥)) = 𝑥)
11		simpl3 1194	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐴 ∈ (SubMnd‘𝑀))
12		wrdf 14459	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Word 𝐽 → 𝑥:(0..^(♯‘𝑥))⟶𝐽)
13	12	ad2antll 729	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥:(0..^(♯‘𝑥))⟶𝐽)
14	13	frnd 6678	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ran 𝑥 ⊆ 𝐽)
15		cores 6210	. . . . . . . . 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 18767	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶Word 𝐼)
18	17	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝑈:𝐼⟶Word 𝐼)
19	18	ffnd 6671	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝑈 Fn 𝐼)
20		fnssres 6623	. . . . . . . . . . 11 ⊢ ((𝑈 Fn 𝐼 ∧ 𝐽 ⊆ 𝐼) → (𝑈 ↾ 𝐽) Fn 𝐽)
21	19, 2, 20	syl2an2r 685	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ↾ 𝐽) Fn 𝐽)
22		df-ima 5644	. . . . . . . . . . 11 ⊢ (𝑈 “ 𝐽) = ran (𝑈 ↾ 𝐽)
23		simprl 770	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 “ 𝐽) ⊆ 𝐴)
24	22, 23	eqsstrrid 3983	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ran (𝑈 ↾ 𝐽) ⊆ 𝐴)
25		df-f 6503	. . . . . . . . . 10 ⊢ ((𝑈 ↾ 𝐽):𝐽⟶𝐴 ↔ ((𝑈 ↾ 𝐽) Fn 𝐽 ∧ ran (𝑈 ↾ 𝐽) ⊆ 𝐴))
26	21, 24, 25	sylanbrc 583	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ↾ 𝐽):𝐽⟶𝐴)
27		wrdco 14773	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝐽 ∧ (𝑈 ↾ 𝐽):𝐽⟶𝐴) → ((𝑈 ↾ 𝐽) ∘ 𝑥) ∈ Word 𝐴)
28	5, 26, 27	syl2anc 584	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ((𝑈 ↾ 𝐽) ∘ 𝑥) ∈ Word 𝐴)
29	16, 28	eqeltrrd 2829	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ∘ 𝑥) ∈ Word 𝐴)
30		gsumwsubmcl 18746	. . . . . . 7 ⊢ ((𝐴 ∈ (SubMnd‘𝑀) ∧ (𝑈 ∘ 𝑥) ∈ Word 𝐴) → (𝑀 Σg (𝑈 ∘ 𝑥)) ∈ 𝐴)
31	11, 29, 30	syl2anc 584	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑀 Σg (𝑈 ∘ 𝑥)) ∈ 𝐴)
32	10, 31	eqeltrrd 2829	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥 ∈ 𝐴)
33	32	expr 456	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ (𝑈 “ 𝐽) ⊆ 𝐴) → (𝑥 ∈ Word 𝐽 → 𝑥 ∈ 𝐴))
34	33	ssrdv 3949	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ (𝑈 “ 𝐽) ⊆ 𝐴) → Word 𝐽 ⊆ 𝐴)
35	34	ex 412	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ 𝐴 → Word 𝐽 ⊆ 𝐴))
36		simpl1 1192	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝐼 ∈ 𝑉)
37		simp2 1137	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝐽 ⊆ 𝐼)
38	37	sselda 3943	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐼)
39	8	vrmdval 18766	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼) → (𝑈‘𝑥) = ⟨“𝑥”⟩)
40	36, 38, 39	syl2anc 584	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → (𝑈‘𝑥) = ⟨“𝑥”⟩)
41		simpr 484	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
42	41	s1cld 14544	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → ⟨“𝑥”⟩ ∈ Word 𝐽)
43	40, 42	eqeltrd 2828	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → (𝑈‘𝑥) ∈ Word 𝐽)
44	43	ralrimiva 3125	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ∀𝑥 ∈ 𝐽 (𝑈‘𝑥) ∈ Word 𝐽)
45	18	ffund 6674	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → Fun 𝑈)
46	18	fdmd 6680	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → dom 𝑈 = 𝐼)
47	37, 46	sseqtrrd 3981	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝐽 ⊆ dom 𝑈)
48		funimass4 6907	. . . . 5 ⊢ ((Fun 𝑈 ∧ 𝐽 ⊆ dom 𝑈) → ((𝑈 “ 𝐽) ⊆ Word 𝐽 ↔ ∀𝑥 ∈ 𝐽 (𝑈‘𝑥) ∈ Word 𝐽))
49	45, 47, 48	syl2anc 584	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ Word 𝐽 ↔ ∀𝑥 ∈ 𝐽 (𝑈‘𝑥) ∈ Word 𝐽))
50	44, 49	mpbird 257	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → (𝑈 “ 𝐽) ⊆ Word 𝐽)
51		sstr2 3950	. . 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 212	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ 𝐴 ↔ Word 𝐽 ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3911  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634   ∘ ccom 5635  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  0cc0 11044  ..^cfzo 13591  ♯chash 14271  Word cword 14454  ⟨“cs1 14536   Σ_g cgsu 17379  SubMndcsubmnd 18691  freeMndcfrmd 18756  var_FMndcvrmd 18757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-n0 12419  df-xnn0 12492  df-z 12506  df-uz 12770  df-fz 13445  df-fzo 13592  df-seq 13943  df-hash 14272  df-word 14455  df-lsw 14504  df-concat 14512  df-s1 14537  df-substr 14582  df-pfx 14612  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-0g 17380  df-gsum 17381  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-submnd 18693  df-frmd 18758  df-vrmd 18759

This theorem is referenced by: (None)