Theorem mgpress 13726

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 2205	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
3	1, 2	mgpvalg 13718	. . . 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 5961	. 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 13720	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑀 ∈ V)
7		ressvalsets 12929	. . . 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 2205	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
10	1, 9	mgpbasg 13721	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑀))
11	10	adantr 276	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (Base‘𝑅) = (Base‘𝑀))
12	11	ineq2d 3374	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐴 ∩ (Base‘𝑅)) = (𝐴 ∩ (Base‘𝑀)))
13	12	opeq2d 3826	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩ = ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑀))⟩)
14	13	oveq2d 5962	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩) = (𝑀 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑀))⟩))
15	8, 14	eqtr4d 2241	. 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 12929	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑅 ↾s 𝐴) = (𝑅 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩))
18	16, 17	eqtrid 2250	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑆 = (𝑅 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩))
19	16, 2	ressmulrg 13010	. . . . . . 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 2211	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (.r‘𝑆) = (.r‘𝑅))
22	21	opeq2d 3826	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ⟨(+g‘ndx), (.r‘𝑆)⟩ = ⟨(+g‘ndx), (.r‘𝑅)⟩)
23	18, 22	oveq12d 5964	. . 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 12930	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑅 ↾s 𝐴) ∈ V)
25	16, 24	eqeltrid 2292	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑆 ∈ V)
26		eqid 2205	. . . . 5 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
27		eqid 2205	. . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆)
28	26, 27	mgpvalg 13718	. . . 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 12977	. . . . . 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 12921	. . . . 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 12990	. . . . . 6 ⊢ (Base‘ndx) ≠ (+g‘ndx)
37	36	necomi 2461	. . . . 5 ⊢ (+g‘ndx) ≠ (Base‘ndx)
38	37	a1i 9	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (+g‘ndx) ≠ (Base‘ndx))
39		mulrslid 12997	. . . . . 6 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
40	39	slotex 12892	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V)
41	40	adantr 276	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (.r‘𝑅) ∈ V)
42		inex1g 4181	. . . . 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 12906	. . 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 2248	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (mulGrp‘𝑆) = ((𝑅 sSet ⟨(+g‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑅))⟩))
46	5, 15, 45	3eqtr4d 2248	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2176   ≠ wne 2376  Vcvv 2772   ∩ cin 3165  ⟨cop 3636  ‘cfv 5272  (class class class)co 5946  ℕcn 9038  ndxcnx 12862   sSet csts 12863  Slot cslot 12864  Basecbs 12865   ↾_s cress 12866  +_gcplusg 12942  ._rcmulr 12943  mulGrpcmgp 13715

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-pre-ltirr 8039  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-ltxr 8114  df-inn 9039  df-2 9097  df-3 9098  df-ndx 12868  df-slot 12869  df-base 12871  df-sets 12872  df-iress 12873  df-plusg 12955  df-mulr 12956  df-mgp 13716

This theorem is referenced by:  rdivmuldivd  13939  subrgcrng  14020  subrgsubm  14029  resrhm  14043  resrhm2b  14044  zringmpg  14401