Theorem subsubm 18750

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 2730	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
2	1	submss 18743	. . . . . . 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 18748	. . . . . . 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 3987	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ 𝑆)
8		eqid 2730	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
9	8	submss 18743	. . . . . 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 3960	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12		eqid 2730	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
13	4, 12	subm0 18749	. . . . . 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 2730	. . . . . . 7 ⊢ (0g‘𝐻) = (0g‘𝐻)
16	15	subm0cl 18745	. . . . . 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 2829	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g‘𝐺) ∈ 𝐴)
19	4	oveq1i 7400	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = ((𝐺 ↾s 𝑆) ↾s 𝐴)
20		ressabs 17225	. . . . . . 7 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝐴) = (𝐺 ↾s 𝐴))
21	19, 20	eqtrid 2777	. . . . . 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 2730	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = (𝐻 ↾s 𝐴)
24	23	submmnd 18747	. . . . . 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 2830	. . . 4 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐺 ↾s 𝐴) ∈ Mnd)
27		submrcl 18736	. . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
28	27	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐺 ∈ Mnd)
29		eqid 2730	. . . . . 6 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴)
30	8, 12, 29	issubm2 18738	. . . . 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 3986	. . 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 18745	. . . . 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 2830	. . 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 18747	. . . . 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 2829	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) ∈ Mnd)
45	4	submmnd 18747	. . . . 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 18738	. . . 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 1540   ∈ wcel 2109   ⊆ wss 3917  ‘cfv 6514  (class class class)co 7390  Basecbs 17186   ↾_s cress 17207  0_gc0g 17409  Mndcmnd 18668  SubMndcsubmnd 18716

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718

This theorem is referenced by:  zrhpsgnmhm  21500  amgmlem  26907  nn0archi  33325  amgmwlem  49795  amgmlemALT  49796