Theorem rhmimasubrnglem 20541

Description: Lemma for rhmimasubrng 20542: Modified part of mhmima 18788. (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 773	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝐹 ∈ (𝑀 MndHom 𝑁))
2		eqid 2741	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	2	subrngss 20524	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (SubRng‘𝑅) → 𝑋 ⊆ (Base‘𝑅))
4		rhmimasubrnglem.b	. . . . . . . . . . . . 13 ⊢ 𝑀 = (mulGrp‘𝑅)
5	4, 2	mgpbas 20121	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑀)
6	3, 5	sseqtrdi 3957	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (SubRng‘𝑅) → 𝑋 ⊆ (Base‘𝑀))
7	6	adantl 483	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → 𝑋 ⊆ (Base‘𝑀))
8	7	adantr 482	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀))
9		simprl 777	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
10	8, 9	sseldd 3918	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ (Base‘𝑀))
11		simprr 779	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
12	8, 11	sseldd 3918	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
13		eqid 2741	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
14		eqid 2741	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
15		eqid 2741	. . . . . . . . 9 ⊢ (+g‘𝑁) = (+g‘𝑁)
16	13, 14, 15	mhmlin 18756	. . . . . . . 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 1380	. . . . . . 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 2741	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
19	13, 18	mhmf 18752	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
20	19	adantr 482	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
21	20	ffnd 6660	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → 𝐹 Fn (Base‘𝑀))
22	21	adantr 482	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝐹 Fn (Base‘𝑀))
23		eqid 2741	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
24	4, 23	mgpplusg 20120	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (+g‘𝑀)
25	24	eqcomi 2750	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (.r‘𝑅)
26	25	subrngmcl 20533	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (SubRng‘𝑅) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
27	26	3expb 1127	. . . . . . . . 9 ⊢ ((𝑋 ∈ (SubRng‘𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
28	27	adantll 721	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
29		fnfvima 7181	. . . . . . . 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 1380	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) ∈ (𝐹 “ 𝑋))
31	17, 30	eqeltrrd 2842	. . . . . 6 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
32	31	anassrs 469	. . . . 5 ⊢ ((((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
33	32	ralrimiva 3133	. . . 4 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
34		oveq2 7368	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧)(+g‘𝑁)𝑦) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
35	34	eleq1d 2826	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
36	35	ralima 7185	. . . . . 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 591	. . . . 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 482	. . . 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 259	. . 3 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧 ∈ 𝑋) → ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
40	39	ralrimiva 3133	. 2 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
41		oveq1 7367	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝑁)𝑦) = ((𝐹‘𝑧)(+g‘𝑁)𝑦))
42	41	eleq1d 2826	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)))
43	42	ralbidv 3164	. . . 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 7185	. . 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 591	. 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 259	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 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055   ⊆ wss 3885   “ cima 5624   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  +_gcplusg 17215  ._rcmulr 17216   MndHom cmhm 18744  mulGrpcmgp 20116  SubRngcsubrng 20521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-mgm 18603  df-sgrp 18682  df-mhm 18746  df-subg 19094  df-abl 19753  df-mgp 20117  df-rng 20129  df-subrng 20522

This theorem is referenced by: (None)