Theorem subsubmgm 43372

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 2772	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
2	1	submgmss 43367	. . . . . . 7 ⊢ (𝐴 ∈ (SubMgm‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
3	2	adantl 474	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4		subsubmgm.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
5	4	submgmbas 43371	. . . . . . 7 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 = (Base‘𝐻))
6	5	adantr 473	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 = (Base‘𝐻))
7	3, 6	sseqtr4d 3894	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ 𝑆)
8		eqid 2772	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
9	8	submgmss 43367	. . . . . 6 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
10	9	adantr 473	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
11	7, 10	sstrd 3864	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12	4	oveq1i 6980	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = ((𝐺 ↾s 𝑆) ↾s 𝐴)
13		ressabs 16409	. . . . . . 7 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝐴) = (𝐺 ↾s 𝐴))
14	12, 13	syl5eq 2820	. . . . . 6 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
15	7, 14	syldan 582	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
16		eqid 2772	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = (𝐻 ↾s 𝐴)
17	16	submgmmgm 43370	. . . . . 6 ⊢ (𝐴 ∈ (SubMgm‘𝐻) → (𝐻 ↾s 𝐴) ∈ Mgm)
18	17	adantl 474	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻 ↾s 𝐴) ∈ Mgm)
19	15, 18	eqeltrrd 2861	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐺 ↾s 𝐴) ∈ Mgm)
20		submgmrcl 43357	. . . . . 6 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝐺 ∈ Mgm)
21	20	adantr 473	. . . . 5 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐺 ∈ Mgm)
22		eqid 2772	. . . . . 6 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴)
23	8, 22	issubmgm2 43365	. . . . 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 700	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ∈ (SubMgm‘𝐺))
26	25, 7	jca 504	. 2 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆))
27		simprr 760	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
28	5	adantr 473	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝑆 = (Base‘𝐻))
29	27, 28	sseqtrd 3893	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ (Base‘𝐻))
30	14	adantrl 703	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
31	22	submgmmgm 43370	. . . . 5 ⊢ (𝐴 ∈ (SubMgm‘𝐺) → (𝐺 ↾s 𝐴) ∈ Mgm)
32	31	ad2antrl 715	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐺 ↾s 𝐴) ∈ Mgm)
33	30, 32	eqeltrd 2860	. . 3 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) ∈ Mgm)
34	4	submgmmgm 43370	. . . . 5 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → 𝐻 ∈ Mgm)
35	34	adantr 473	. . . 4 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐻 ∈ Mgm)
36	1, 16	issubmgm2 43365	. . . 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 700	. 2 ⊢ ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ∈ (SubMgm‘𝐻))
39	26, 38	impbida 788	1 ⊢ (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2048   ⊆ wss 3825  ‘cfv 6182  (class class class)co 6970  Basecbs 16329   ↾_s cress 16330  Mgmcmgm 17698  SubMgmcsubmgm 43353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-ndx 16332  df-slot 16333  df-base 16335  df-sets 16336  df-ress 16337  df-plusg 16424  df-mgm 17700  df-submgm 43355

This theorem is referenced by: (None)