Theorem subsubmgm 44777

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 2759	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
2	1	submgmss 44772	. . . . . . 7 ⊢ (𝐴 ∈ (SubMgm‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
3	2	adantl 486	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4		subsubmgm.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
5	4	submgmbas 44776	. . . . . . 7 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 = (Base‘𝐻))
6	5	adantr 485	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 = (Base‘𝐻))
7	3, 6	sseqtrrd 3934	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ 𝑆)
8		eqid 2759	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
9	8	submgmss 44772	. . . . . 6 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	9	adantr 485	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
11	7, 10	sstrd 3903	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12	4	oveq1i 7161	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = ((𝐺 ↾s 𝑆) ↾s 𝐴)
13		ressabs 16614	. . . . . . 7 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝐴) = (𝐺 ↾s 𝐴))
14	12, 13	syl5eq 2806	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
15	7, 14	syldan 595	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
16		eqid 2759	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = (𝐻 ↾s 𝐴)
17	16	submgmmgm 44775	. . . . . 6 ⊢ (𝐴 ∈ (SubMgm‘𝐻) → (𝐻 ↾s 𝐴) ∈ Mgm)
18	17	adantl 486	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻 ↾s 𝐴) ∈ Mgm)
19	15, 18	eqeltrrd 2854	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐺 ↾s 𝐴) ∈ Mgm)
20		submgmrcl 44762	. . . . . 6 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝐺 ∈ Mgm)
21	20	adantr 485	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐺 ∈ Mgm)
22		eqid 2759	. . . . . 6 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴)
23	8, 22	issubmgm2 44770	. . . . 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 516	. 2 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆))
27		simprr 773	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
28	5	adantr 485	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝑆 = (Base‘𝐻))
29	27, 28	sseqtrd 3933	. . 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 44775	. . . . 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 2853	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) ∈ Mgm)
34	4	submgmmgm 44775	. . . . 5 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝐻 ∈ Mgm)
35	34	adantr 485	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐻 ∈ Mgm)
36	1, 16	issubmgm2 44770	. . . 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 801	1 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ⊆ wss 3859  ‘cfv 6336  (class class class)co 7151  Basecbs 16534   ↾_s cress 16535  Mgmcmgm 17909  SubMgmcsubmgm 44758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10624  ax-resscn 10625  ax-1cn 10626  ax-icn 10627  ax-addcl 10628  ax-addrcl 10629  ax-mulcl 10630  ax-mulrcl 10631  ax-mulcom 10632  ax-addass 10633  ax-mulass 10634  ax-distr 10635  ax-i2m1 10636  ax-1ne0 10637  ax-1rid 10638  ax-rnegex 10639  ax-rrecex 10640  ax-cnre 10641  ax-pre-lttri 10642  ax-pre-lttrn 10643  ax-pre-ltadd 10644  ax-pre-mulgt0 10645

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-pnf 10708  df-mnf 10709  df-xr 10710  df-ltxr 10711  df-le 10712  df-sub 10903  df-neg 10904  df-nn 11668  df-2 11730  df-ndx 16537  df-slot 16538  df-base 16540  df-sets 16541  df-ress 16542  df-plusg 16629  df-mgm 17911  df-submgm 44760

This theorem is referenced by: (None)