Theorem subsubm 18719

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 18712	. . . . . . 7 ⊢ (𝐴 ∈ (SubMnd‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
3	2	adantl 481	. . . . . 6 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4		subsubm.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
5	4	submbas 18717	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘𝐻))
6	5	adantr 480	. . . . . 6 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 = (Base‘𝐻))
7	3, 6	sseqtrrd 3967	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ 𝑆)
8		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
9	8	submss 18712	. . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	9	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
11	7, 10	sstrd 3940	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12		eqid 2731	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	4, 12	subm0 18718	. . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
14	13	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g‘𝐺) = (0g‘𝐻))
15		eqid 2731	. . . . . . 7 ⊢ (0g‘𝐻) = (0g‘𝐻)
16	15	subm0cl 18714	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝐻) → (0g‘𝐻) ∈ 𝐴)
17	16	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g‘𝐻) ∈ 𝐴)
18	14, 17	eqeltrd 2831	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g‘𝐺) ∈ 𝐴)
19	4	oveq1i 7351	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = ((𝐺 ↾s 𝑆) ↾s 𝐴)
20		ressabs 17154	. . . . . . 7 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝐴) = (𝐺 ↾s 𝐴))
21	19, 20	eqtrid 2778	. . . . . 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 18716	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝐻) → (𝐻 ↾s 𝐴) ∈ Mnd)
25	24	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻 ↾s 𝐴) ∈ Mnd)
26	22, 25	eqeltrrd 2832	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐺 ↾s 𝐴) ∈ Mnd)
27		submrcl 18705	. . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
28	27	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐺 ∈ Mnd)
29		eqid 2731	. . . . . 6 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴)
30	8, 12, 29	issubm2 18707	. . . . 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 1343	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ∈ (SubMnd‘𝐺))
33	32, 7	jca 511	. 2 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆))
34		simprr 772	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
35	5	adantr 480	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝑆 = (Base‘𝐻))
36	34, 35	sseqtrd 3966	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ (Base‘𝐻))
37	13	adantr 480	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (0g‘𝐺) = (0g‘𝐻))
38	12	subm0cl 18714	. . . . 5 ⊢ (𝐴 ∈ (SubMnd‘𝐺) → (0g‘𝐺) ∈ 𝐴)
39	38	ad2antrl 728	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (0g‘𝐺) ∈ 𝐴)
40	37, 39	eqeltrrd 2832	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (0g‘𝐻) ∈ 𝐴)
41	21	adantrl 716	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
42	29	submmnd 18716	. . . . 5 ⊢ (𝐴 ∈ (SubMnd‘𝐺) → (𝐺 ↾s 𝐴) ∈ Mnd)
43	42	ad2antrl 728	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐺 ↾s 𝐴) ∈ Mnd)
44	41, 43	eqeltrd 2831	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) ∈ Mnd)
45	4	submmnd 18716	. . . . 5 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐻 ∈ Mnd)
46	45	adantr 480	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐻 ∈ Mnd)
47	1, 15, 23	issubm2 18707	. . . 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 1343	. 2 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ∈ (SubMnd‘𝐻))
50	33, 49	impbida 800	1 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ‘cfv 6476  (class class class)co 7341  Basecbs 17115   ↾_s cress 17136  0_gc0g 17338  Mndcmnd 18637  SubMndcsubmnd 18685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-0g 17340  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-submnd 18687

This theorem is referenced by:  zrhpsgnmhm  21516  amgmlem  26922  nn0archi  33304  amgmwlem  49834  amgmlemALT  49835