Theorem frmdss2 18673

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 1191	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐼 ∈ 𝑉)
2		simpl2 1192	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐽 ⊆ 𝐼)
3		sswrd 14410	. . . . . . . . 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 771	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥 ∈ Word 𝐽)
6	4, 5	sseldd 3945	. . . . . . 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 18672	. . . . . . 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 1193	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝐴 ∈ (SubMnd‘𝑀))
12		wrdf 14407	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Word 𝐽 → 𝑥:(0..^(♯‘𝑥))⟶𝐽)
13	12	ad2antll 727	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → 𝑥:(0..^(♯‘𝑥))⟶𝐽)
14	13	frnd 6676	. . . . . . . . 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 18668	. . . . . . . . . . . . 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 6669	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → 𝑈 Fn 𝐼)
20		fnssres 6624	. . . . . . . . . . 11 ⊢ ((𝑈 Fn 𝐼 ∧ 𝐽 ⊆ 𝐼) → (𝑈 ↾ 𝐽) Fn 𝐽)
21	19, 2, 20	syl2an2r 683	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ↾ 𝐽) Fn 𝐽)
22		df-ima 5646	. . . . . . . . . . 11 ⊢ (𝑈 “ 𝐽) = ran (𝑈 ↾ 𝐽)
23		simprl 769	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 “ 𝐽) ⊆ 𝐴)
24	22, 23	eqsstrrid 3993	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → ran (𝑈 ↾ 𝐽) ⊆ 𝐴)
25		df-f 6500	. . . . . . . . . 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 14720	. . . . . . . . 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 2839	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ ((𝑈 “ 𝐽) ⊆ 𝐴 ∧ 𝑥 ∈ Word 𝐽)) → (𝑈 ∘ 𝑥) ∈ Word 𝐴)
30		gsumwsubmcl 18647	. . . . . . 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 2839	. . . . 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 3950	. . 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 1191	. . . . . . 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 3944	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐼)
39	8	vrmdval 18667	. . . . . . 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 485	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽)
42	41	s1cld 14491	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → ⟨“𝑥”⟩ ∈ Word 𝐽)
43	40, 42	eqeltrd 2838	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) ∧ 𝑥 ∈ 𝐽) → (𝑈‘𝑥) ∈ Word 𝐽)
44	43	ralrimiva 3143	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ∀𝑥 ∈ 𝐽 (𝑈‘𝑥) ∈ Word 𝐽)
45	18	ffund 6672	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → Fun 𝑈)
46	18	fdmd 6679	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → dom 𝑈 = 𝐼)
47	37, 46	sseqtrrd 3985	. . . . 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 256	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → (𝑈 “ 𝐽) ⊆ Word 𝐽)
51		sstr2 3951	. . 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 211	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼 ∧ 𝐴 ∈ (SubMnd‘𝑀)) → ((𝑈 “ 𝐽) ⊆ 𝐴 ↔ Word 𝐽 ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3910  dom cdm 5633  ran crn 5634   ↾ cres 5635   “ cima 5636   ∘ ccom 5637  Fun wfun 6490   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  0cc0 11051  ..^cfzo 13567  ♯chash 14230  Word cword 14402  ⟨“cs1 14483   Σ_g cgsu 17322  SubMndcsubmnd 18600  freeMndcfrmd 18657  var_FMndcvrmd 18658

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-rmo 3353  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-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-frmd 18659  df-vrmd 18660

This theorem is referenced by: (None)