Theorem rhmimasubrnglem 20474

Description: Lemma for rhmimasubrng 20475: Modified part of mhmima 18752. (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 20457	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (SubRng‘𝑅) → 𝑋 ⊆ (Base‘𝑅))
4		rhmimasubrnglem.b	. . . . . . . . . . . . 13 ⊢ 𝑀 = (mulGrp‘𝑅)
5	4, 2	mgpbas 20054	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑀)
6	3, 5	sseqtrdi 3987	. . . . . . . . . . 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 3947	. . . . . . . 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 3947	. . . . . . . 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 18720	. . . . . . . 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 18716	. . . . . . . . . . 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 6689	. . . . . . . . 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 20053	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (+g‘𝑀)
25	24	eqcomi 2738	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (.r‘𝑅)
26	25	subrngmcl 20466	. . . . . . . . . 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 7207	. . . . . . . 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 3125	. . . 4 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
34		oveq2 7395	. . . . . . . 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 7211	. . . . . 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 3125	. 2 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
41		oveq1 7394	. . . . . 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 3156	. . . 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 7211	. . 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 3914   “ cima 5641   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221   MndHom cmhm 18708  mulGrpcmgp 20049  SubRngcsubrng 20454

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 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-mgm 18567  df-sgrp 18646  df-mhm 18710  df-subg 19055  df-abl 19713  df-mgp 20050  df-rng 20062  df-subrng 20455

This theorem is referenced by: (None)