Theorem rhmimasubrnglem 20450

Description: Lemma for rhmimasubrng 20451: Modified part of mhmima 18699. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 16-Feb-2025.)

Hypothesis

Ref	Expression
rhmimasubrnglem.b	⊢ 𝑀 = (mulGrp‘𝑅)

Assertion

Ref	Expression
rhmimasubrnglem	⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑀   𝑥,𝑁,𝑦   𝑥,𝑅   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑅(𝑦)   𝑀(𝑦)

Proof of Theorem rhmimasubrnglem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 766	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝐹 ∈ (𝑀 MndHom 𝑁))
2		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	2	subrngss 20433	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (SubRng‘𝑅) → 𝑋 ⊆ (Base‘𝑅))
4		rhmimasubrnglem.b	. . . . . . . . . . . . 13 ⊢ 𝑀 = (mulGrp‘𝑅)
5	4, 2	mgpbas 20030	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑀)
6	3, 5	sseqtrdi 3976	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (SubRng‘𝑅) → 𝑋 ⊆ (Base‘𝑀))
7	6	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → 𝑋 ⊆ (Base‘𝑀))
8	7	adantr 480	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀))
9		simprl 770	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
10	8, 9	sseldd 3936	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ (Base‘𝑀))
11		simprr 772	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
12	8, 11	sseldd 3936	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
13		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
14		eqid 2729	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
15		eqid 2729	. . . . . . . . 9 ⊢ (+g‘𝑁) = (+g‘𝑁)
16	13, 14, 15	mhmlin 18667	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑧 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
17	1, 10, 12, 16	syl3anc 1373	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
18		eqid 2729	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
19	13, 18	mhmf 18663	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
20	19	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
21	20	ffnd 6653	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → 𝐹 Fn (Base‘𝑀))
22	21	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝐹 Fn (Base‘𝑀))
23		eqid 2729	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
24	4, 23	mgpplusg 20029	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (+g‘𝑀)
25	24	eqcomi 2738	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (.r‘𝑅)
26	25	subrngmcl 20442	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (SubRng‘𝑅) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
27	26	3expb 1120	. . . . . . . . 9 ⊢ ((𝑋 ∈ (SubRng‘𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
28	27	adantll 714	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
29		fnfvima 7169	. . . . . . . 8 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (𝑧(+g‘𝑀)𝑥) ∈ 𝑋) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) ∈ (𝐹 “ 𝑋))
30	22, 8, 28, 29	syl3anc 1373	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) ∈ (𝐹 “ 𝑋))
31	17, 30	eqeltrrd 2829	. . . . . 6 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
32	31	anassrs 467	. . . . 5 ⊢ ((((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
33	32	ralrimiva 3121	. . . 4 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
34		oveq2 7357	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧)(+g‘𝑁)𝑦) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
35	34	eleq1d 2813	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
36	35	ralima 7173	. . . . . 6 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
37	21, 7, 36	syl2anc 584	. . . . 5 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
38	37	adantr 480	. . . 4 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧 ∈ 𝑋) → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
39	33, 38	mpbird 257	. . 3 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧 ∈ 𝑋) → ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
40	39	ralrimiva 3121	. 2 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
41		oveq1 7356	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝑁)𝑦) = ((𝐹‘𝑧)(+g‘𝑁)𝑦))
42	41	eleq1d 2813	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)))
43	42	ralbidv 3152	. . . 4 ⊢ (𝑥 = (𝐹‘𝑧) → (∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)))
44	43	ralima 7173	. . 3 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)))
45	21, 7, 44	syl2anc 584	. 2 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → (∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)))
46	40, 45	mpbird 257	1 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903   “ cima 5622   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162   MndHom cmhm 18655  mulGrpcmgp 20025  SubRngcsubrng 20430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-mgm 18514  df-sgrp 18593  df-mhm 18657  df-subg 19002  df-abl 19662  df-mgp 20026  df-rng 20038  df-subrng 20431

This theorem is referenced by: (None)