Theorem subsubm 18673

Description: A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)

Hypothesis

Ref	Expression
subsubm.h	⊢ 𝐻 = (𝐺 ↾s 𝑆)

Assertion

Ref	Expression
subsubm	⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Proof of Theorem subsubm

Step	Hyp	Ref	Expression
1		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
2	1	submss 18666	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
3	2	adantl 482	. . . . . 6 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4		subsubm.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
5	4	submbas 18671	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘𝐻))
6	5	adantr 481	. . . . . 6 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 = (Base‘𝐻))
7	3, 6	sseqtrrd 4019	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ 𝑆)
8		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
9	8	submss 18666	. . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	9	adantr 481	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
11	7, 10	sstrd 3988	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12		eqid 2731	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	4, 12	subm0 18672	. . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
14	13	adantr 481	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g‘𝐺) = (0g‘𝐻))
15		eqid 2731	. . . . . . 7 ⊢ (0g‘𝐻) = (0g‘𝐻)
16	15	subm0cl 18668	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝐻) → (0g‘𝐻) ∈ 𝐴)
17	16	adantl 482	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g‘𝐻) ∈ 𝐴)
18	14, 17	eqeltrd 2832	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g‘𝐺) ∈ 𝐴)
19	4	oveq1i 7403	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = ((𝐺 ↾s 𝑆) ↾s 𝐴)
20		ressabs 17176	. . . . . . 7 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝐴) = (𝐺 ↾s 𝐴))
21	19, 20	eqtrid 2783	. . . . . 6 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
22	7, 21	syldan 591	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
23		eqid 2731	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = (𝐻 ↾s 𝐴)
24	23	submmnd 18670	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝐻) → (𝐻 ↾s 𝐴) ∈ Mnd)
25	24	adantl 482	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻 ↾s 𝐴) ∈ Mnd)
26	22, 25	eqeltrrd 2833	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐺 ↾s 𝐴) ∈ Mnd)
27		submrcl 18659	. . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
28	27	adantr 481	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐺 ∈ Mnd)
29		eqid 2731	. . . . . 6 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴)
30	8, 12, 29	issubm2 18661	. . . . 5 ⊢ (𝐺 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝐺) ↔ (𝐴 ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ 𝐴 ∧ (𝐺 ↾s 𝐴) ∈ Mnd)))
31	28, 30	syl 17	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐴 ∈ (SubMnd‘𝐺) ↔ (𝐴 ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ 𝐴 ∧ (𝐺 ↾s 𝐴) ∈ Mnd)))
32	11, 18, 26, 31	mpbir3and 1342	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ∈ (SubMnd‘𝐺))
33	32, 7	jca 512	. 2 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆))
34		simprr 771	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
35	5	adantr 481	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝑆 = (Base‘𝐻))
36	34, 35	sseqtrd 4018	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ (Base‘𝐻))
37	13	adantr 481	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (0g‘𝐺) = (0g‘𝐻))
38	12	subm0cl 18668	. . . . 5 ⊢ (𝐴 ∈ (SubMnd‘𝐺) → (0g‘𝐺) ∈ 𝐴)
39	38	ad2antrl 726	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (0g‘𝐺) ∈ 𝐴)
40	37, 39	eqeltrrd 2833	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (0g‘𝐻) ∈ 𝐴)
41	21	adantrl 714	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
42	29	submmnd 18670	. . . . 5 ⊢ (𝐴 ∈ (SubMnd‘𝐺) → (𝐺 ↾s 𝐴) ∈ Mnd)
43	42	ad2antrl 726	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐺 ↾s 𝐴) ∈ Mnd)
44	41, 43	eqeltrd 2832	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) ∈ Mnd)
45	4	submmnd 18670	. . . . 5 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐻 ∈ Mnd)
46	45	adantr 481	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐻 ∈ Mnd)
47	1, 15, 23	issubm2 18661	. . . 4 ⊢ (𝐻 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ⊆ (Base‘𝐻) ∧ (0g‘𝐻) ∈ 𝐴 ∧ (𝐻 ↾s 𝐴) ∈ Mnd)))
48	46, 47	syl 17	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ⊆ (Base‘𝐻) ∧ (0g‘𝐻) ∈ 𝐴 ∧ (𝐻 ↾s 𝐴) ∈ Mnd)))
49	36, 40, 44, 48	mpbir3and 1342	. 2 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ∈ (SubMnd‘𝐻))
50	33, 49	impbida 799	1 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ⊆ wss 3944  ‘cfv 6532  (class class class)co 7393  Basecbs 17126   ↾_s cress 17155  0_gc0g 17367  Mndcmnd 18602  SubMndcsubmnd 18646

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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-0g 17369  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-submnd 18648

This theorem is referenced by:  zrhpsgnmhm  21070  amgmlem  26421  nn0archi  32324  amgmwlem  47497  amgmlemALT  47498