Theorem subsubmgm 18673

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 2725	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
2	1	submgmss 18668	. . . . . . 7 ⊢ (𝐴 ∈ (SubMgm‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
3	2	adantl 480	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4		subsubmgm.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
5	4	submgmbas 18672	. . . . . . 7 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 = (Base‘𝐻))
6	5	adantr 479	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 = (Base‘𝐻))
7	3, 6	sseqtrrd 4018	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ 𝑆)
8		eqid 2725	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
9	8	submgmss 18668	. . . . . 6 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	9	adantr 479	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
11	7, 10	sstrd 3987	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12	4	oveq1i 7429	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = ((𝐺 ↾s 𝑆) ↾s 𝐴)
13		ressabs 17233	. . . . . . 7 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝐴) = (𝐺 ↾s 𝐴))
14	12, 13	eqtrid 2777	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
15	7, 14	syldan 589	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
16		eqid 2725	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = (𝐻 ↾s 𝐴)
17	16	submgmmgm 18671	. . . . . 6 ⊢ (𝐴 ∈ (SubMgm‘𝐻) → (𝐻 ↾s 𝐴) ∈ Mgm)
18	17	adantl 480	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻 ↾s 𝐴) ∈ Mgm)
19	15, 18	eqeltrrd 2826	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐺 ↾s 𝐴) ∈ Mgm)
20		submgmrcl 18658	. . . . . 6 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝐺 ∈ Mgm)
21	20	adantr 479	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐺 ∈ Mgm)
22		eqid 2725	. . . . . 6 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴)
23	8, 22	issubmgm2 18666	. . . . 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 711	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ∈ (SubMgm‘𝐺))
26	25, 7	jca 510	. 2 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆))
27		simprr 771	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
28	5	adantr 479	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝑆 = (Base‘𝐻))
29	27, 28	sseqtrd 4017	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ (Base‘𝐻))
30	14	adantrl 714	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
31	22	submgmmgm 18671	. . . . 5 ⊢ (𝐴 ∈ (SubMgm‘𝐺) → (𝐺 ↾s 𝐴) ∈ Mgm)
32	31	ad2antrl 726	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐺 ↾s 𝐴) ∈ Mgm)
33	30, 32	eqeltrd 2825	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) ∈ Mgm)
34	4	submgmmgm 18671	. . . . 5 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝐻 ∈ Mgm)
35	34	adantr 479	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐻 ∈ Mgm)
36	1, 16	issubmgm2 18666	. . . 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 711	. 2 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ∈ (SubMgm‘𝐻))
39	26, 38	impbida 799	1 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ⊆ wss 3944  ‘cfv 6549  (class class class)co 7419  Basecbs 17183   ↾_s cress 17212  Mgmcmgm 18601  SubMgmcsubmgm 18654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mgm 18603  df-submgm 18656

This theorem is referenced by: (None)