Theorem mgpress 13146

Description: Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)

Hypotheses

Ref	Expression
mgpress.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)
mgpress.2	⊢ 𝑀 = (mulGrp‘𝑅)

Assertion

Ref	Expression
mgpress	⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆))

Proof of Theorem mgpress

Step	Hyp	Ref	Expression
1		mgpress.2	. . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅)
2		eqid 2177	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
3	1, 2	mgpvalg 13138	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑀 = (𝑅 sSet ⟨(+g‘ndx), (.r‘𝑅)⟩))
4	3	adantr 276	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑀 = (𝑅 sSet ⟨(+g‘ndx), (.r‘𝑅)⟩))
5	4	oveq1d 5892	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩) = ((𝑅 sSet ⟨(+g‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩))
6	1	mgpex 13140	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑀 ∈ V)
7		ressvalsets 12526	. . . 4 ⊢ ((𝑀 ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (𝑀 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑀))⟩))
8	6, 7	sylan 283	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (𝑀 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑀))⟩))
9		eqid 2177	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
10	1, 9	mgpbasg 13141	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑀))
11	10	adantr 276	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (Base‘𝑅) = (Base‘𝑀))
12	11	ineq2d 3338	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐴 ∩ (Base‘𝑅)) = (𝐴 ∩ (Base‘𝑀)))
13	12	opeq2d 3787	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑀))⟩)
14	13	oveq2d 5893	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩) = (𝑀 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑀))⟩))
15	8, 14	eqtr4d 2213	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (𝑀 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩))
16		mgpress.1	. . . . 5 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
17		ressvalsets 12526	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑅 ↾s 𝐴) = (𝑅 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩))
18	16, 17	eqtrid 2222	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑆 = (𝑅 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩))
19	16, 2	ressmulrg 12605	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆))
20	19	ancoms 268	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (.r‘𝑅) = (.r‘𝑆))
21	20	eqcomd 2183	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (.r‘𝑆) = (.r‘𝑅))
22	21	opeq2d 3787	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ⟨(+g‘ndx), (.r‘𝑆)⟩ = ⟨(+g‘ndx), (.r‘𝑅)⟩)
23	18, 22	oveq12d 5895	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑆 sSet ⟨(+g‘ndx), (.r‘𝑆)⟩) = ((𝑅 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩) sSet ⟨(+g‘ndx), (.r‘𝑅)⟩))
24		ressex 12527	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑅 ↾s 𝐴) ∈ V)
25	16, 24	eqeltrid 2264	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑆 ∈ V)
26		eqid 2177	. . . . 5 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
27		eqid 2177	. . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆)
28	26, 27	mgpvalg 13138	. . . 4 ⊢ (𝑆 ∈ V → (mulGrp‘𝑆) = (𝑆 sSet ⟨(+g‘ndx), (.r‘𝑆)⟩))
29	25, 28	syl 14	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (mulGrp‘𝑆) = (𝑆 sSet ⟨(+g‘ndx), (.r‘𝑆)⟩))
30		plusgslid 12573	. . . . . 6 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
31	30	simpri 113	. . . . 5 ⊢ (+g‘ndx) ∈ ℕ
32	31	a1i 9	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (+g‘ndx) ∈ ℕ)
33		basendxnn 12520	. . . . 5 ⊢ (Base‘ndx) ∈ ℕ
34	33	a1i 9	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (Base‘ndx) ∈ ℕ)
35		simpl 109	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑅 ∈ 𝑉)
36		basendxnplusgndx 12585	. . . . . 6 ⊢ (Base‘ndx) ≠ (+g‘ndx)
37	36	necomi 2432	. . . . 5 ⊢ (+g‘ndx) ≠ (Base‘ndx)
38	37	a1i 9	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (+g‘ndx) ≠ (Base‘ndx))
39		mulrslid 12592	. . . . . 6 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
40	39	slotex 12491	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V)
41	40	adantr 276	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (.r‘𝑅) ∈ V)
42		inex1g 4141	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ∩ (Base‘𝑅)) ∈ V)
43	42	adantl 277	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐴 ∩ (Base‘𝑅)) ∈ V)
44	32, 34, 35, 38, 41, 43	setscomd 12505	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑅 sSet ⟨(+g‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩) = ((𝑅 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩) sSet ⟨(+g‘ndx), (.r‘𝑅)⟩))
45	23, 29, 44	3eqtr4d 2220	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (mulGrp‘𝑆) = ((𝑅 sSet ⟨(+g‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩))
46	5, 15, 45	3eqtr4d 2220	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  Vcvv 2739   ∩ cin 3130  ⟨cop 3597  ‘cfv 5218  (class class class)co 5877  ℕcn 8921  ndxcnx 12461   sSet csts 12462  Slot cslot 12463  Basecbs 12464   ↾_s cress 12465  +_gcplusg 12538  ._rcmulr 12539  mulGrpcmgp 13135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-mulr 12552  df-mgp 13136

This theorem is referenced by:  rdivmuldivd  13318  subrgcrng  13351  subrgsubm  13360  zringmpg  13581