Theorem subsubmgm 18618

Description: A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.)

Hypothesis

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

Assertion

Ref	Expression
subsubmgm	⊢ (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Proof of Theorem subsubmgm

Step	Hyp	Ref	Expression
1		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
2	1	submgmss 18613	. . . . . . 7 ⊢ (𝐴 ∈ (SubMgm‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
3	2	adantl 481	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4		subsubmgm.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
5	4	submgmbas 18617	. . . . . . 7 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 = (Base‘𝐻))
6	5	adantr 480	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 = (Base‘𝐻))
7	3, 6	sseqtrrd 3967	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ 𝑆)
8		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
9	8	submgmss 18613	. . . . . 6 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	9	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
11	7, 10	sstrd 3940	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12	4	oveq1i 7356	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = ((𝐺 ↾s 𝑆) ↾s 𝐴)
13		ressabs 17159	. . . . . . 7 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝐴) = (𝐺 ↾s 𝐴))
14	12, 13	eqtrid 2778	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
15	7, 14	syldan 591	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
16		eqid 2731	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = (𝐻 ↾s 𝐴)
17	16	submgmmgm 18616	. . . . . 6 ⊢ (𝐴 ∈ (SubMgm‘𝐻) → (𝐻 ↾s 𝐴) ∈ Mgm)
18	17	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻 ↾s 𝐴) ∈ Mgm)
19	15, 18	eqeltrrd 2832	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐺 ↾s 𝐴) ∈ Mgm)
20		submgmrcl 18603	. . . . . 6 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝐺 ∈ Mgm)
21	20	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐺 ∈ Mgm)
22		eqid 2731	. . . . . 6 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴)
23	8, 22	issubmgm2 18611	. . . . 5 ⊢ (𝐺 ∈ Mgm → (𝐴 ∈ (SubMgm‘𝐺) ↔ (𝐴 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝐴) ∈ Mgm)))
24	21, 23	syl 17	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐴 ∈ (SubMgm‘𝐺) ↔ (𝐴 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝐴) ∈ Mgm)))
25	11, 19, 24	mpbir2and 713	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ∈ (SubMgm‘𝐺))
26	25, 7	jca 511	. 2 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆))
27		simprr 772	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
28	5	adantr 480	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝑆 = (Base‘𝐻))
29	27, 28	sseqtrd 3966	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ (Base‘𝐻))
30	14	adantrl 716	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
31	22	submgmmgm 18616	. . . . 5 ⊢ (𝐴 ∈ (SubMgm‘𝐺) → (𝐺 ↾s 𝐴) ∈ Mgm)
32	31	ad2antrl 728	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐺 ↾s 𝐴) ∈ Mgm)
33	30, 32	eqeltrd 2831	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) ∈ Mgm)
34	4	submgmmgm 18616	. . . . 5 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝐻 ∈ Mgm)
35	34	adantr 480	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐻 ∈ Mgm)
36	1, 16	issubmgm2 18611	. . . 4 ⊢ (𝐻 ∈ Mgm → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ⊆ (Base‘𝐻) ∧ (𝐻 ↾s 𝐴) ∈ Mgm)))
37	35, 36	syl 17	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ⊆ (Base‘𝐻) ∧ (𝐻 ↾s 𝐴) ∈ Mgm)))
38	29, 33, 37	mpbir2and 713	. 2 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ∈ (SubMgm‘𝐻))
39	26, 38	impbida 800	1 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ‘cfv 6481  (class class class)co 7346  Basecbs 17120   ↾_s cress 17141  Mgmcmgm 18546  SubMgmcsubmgm 18599

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 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

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-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 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mgm 18548  df-submgm 18601

This theorem is referenced by: (None)