Theorem rngqiprnglin 21263

Description: 𝐹 is linear with respect to the multiplication. (Contributed by AV, 28-Feb-2025.)

Hypotheses

Ref	Expression
rng2idlring.r	⊢ (𝜑 → 𝑅 ∈ Rng)
rng2idlring.i	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
rng2idlring.j	⊢ 𝐽 = (𝑅 ↾s 𝐼)
rng2idlring.u	⊢ (𝜑 → 𝐽 ∈ Ring)
rng2idlring.b	⊢ 𝐵 = (Base‘𝑅)
rng2idlring.t	⊢ · = (.r‘𝑅)
rng2idlring.1	⊢ 1 = (1r‘𝐽)
rngqiprngim.g	⊢ ∼ = (𝑅 ~QG 𝐼)
rngqiprngim.q	⊢ 𝑄 = (𝑅 /s ∼ )
rngqiprngim.c	⊢ 𝐶 = (Base‘𝑄)
rngqiprngim.p	⊢ 𝑃 = (𝑄 ×s 𝐽)
rngqiprngim.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)

Assertion

Ref	Expression
rngqiprnglin	⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(.r‘𝑃)(𝐹‘𝑏)))

Distinct variable groups:   𝑥,𝐶   𝑥,𝐼   𝑥,𝐵   𝜑,𝑥   𝑥, ∼   𝑥, 1   𝑥, ·   𝐵,𝑎,𝑏   𝐹,𝑎,𝑏   𝑃,𝑎,𝑏   𝑅,𝑎,𝑏,𝑥   𝜑,𝑎,𝑏   𝐽,𝑎   𝑄,𝑎   𝐶,𝑎,𝑏   𝐼,𝑎,𝑏   ∼ ,𝑎   1 ,𝑎   · ,𝑎

Allowed substitution hints:   𝑃(𝑥)   𝑄(𝑥,𝑏)   ∼ (𝑏)   · (𝑏)   1 (𝑏)   𝐹(𝑥)   𝐽(𝑥,𝑏)

Proof of Theorem rngqiprnglin

Step	Hyp	Ref	Expression
1		rngqiprngim.p	. . . . 5 ⊢ 𝑃 = (𝑄 ×s 𝐽)
2		eqid 2735	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
3		eqid 2735	. . . . 5 ⊢ (Base‘𝐽) = (Base‘𝐽)
4		rngqiprngim.q	. . . . . . 7 ⊢ 𝑄 = (𝑅 /s ∼ )
5	4	ovexi 7439	. . . . . 6 ⊢ 𝑄 ∈ V
6	5	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑄 ∈ V)
7		rng2idlring.u	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Ring)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝐽 ∈ Ring)
9		rng2idlring.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Rng)
10		simpl 482	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐵)
11		rngqiprngim.g	. . . . . . 7 ⊢ ∼ = (𝑅 ~QG 𝐼)
12		rng2idlring.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
13	11, 4, 12, 2	quseccl0 19168	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵) → [𝑎] ∼ ∈ (Base‘𝑄))
14	9, 10, 13	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑎] ∼ ∈ (Base‘𝑄))
15		rng2idlring.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
16		rng2idlring.j	. . . . . . 7 ⊢ 𝐽 = (𝑅 ↾s 𝐼)
17		rng2idlring.t	. . . . . . 7 ⊢ · = (.r‘𝑅)
18		rng2idlring.1	. . . . . . 7 ⊢ 1 = (1r‘𝐽)
19	9, 15, 16, 7, 12, 17, 18	rngqiprngghmlem1 21248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ( 1 · 𝑎) ∈ (Base‘𝐽))
20	10, 19	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑎) ∈ (Base‘𝐽))
21		simpr 484	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
22	11, 4, 12, 2	quseccl0 19168	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑏 ∈ 𝐵) → [𝑏] ∼ ∈ (Base‘𝑄))
23	9, 21, 22	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑏] ∼ ∈ (Base‘𝑄))
24	9, 15, 16, 7, 12, 17, 18	rngqiprngghmlem1 21248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ( 1 · 𝑏) ∈ (Base‘𝐽))
25	21, 24	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑏) ∈ (Base‘𝐽))
26	9, 15, 16, 7, 12, 17, 18, 11, 4	rngqiprnglinlem3 21254	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (.r‘𝑄)[𝑏] ∼ ) ∈ (Base‘𝑄))
27		eqid 2735	. . . . . 6 ⊢ (.r‘𝐽) = (.r‘𝐽)
28	3, 27, 8, 20, 25	ringcld 20220	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(.r‘𝐽)( 1 · 𝑏)) ∈ (Base‘𝐽))
29		eqid 2735	. . . . 5 ⊢ (.r‘𝑄) = (.r‘𝑄)
30		eqid 2735	. . . . 5 ⊢ (.r‘𝑃) = (.r‘𝑃)
31	1, 2, 3, 6, 8, 14, 20, 23, 25, 26, 28, 29, 27, 30	xpsmul 17589	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩(.r‘𝑃)⟨[𝑏] ∼ , ( 1 · 𝑏)⟩) = ⟨([𝑎] ∼ (.r‘𝑄)[𝑏] ∼ ), (( 1 · 𝑎)(.r‘𝐽)( 1 · 𝑏))⟩)
32	9, 15, 16, 7, 12, 17, 18, 11, 4	rngqiprnglinlem2 21253	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎 · 𝑏)] ∼ = ([𝑎] ∼ (.r‘𝑄)[𝑏] ∼ ))
33	32	eqcomd 2741	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (.r‘𝑄)[𝑏] ∼ ) = [(𝑎 · 𝑏)] ∼ )
34	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝐼 ∈ (2Ideal‘𝑅))
35	16, 17	ressmulr 17321	. . . . . . . . 9 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → · = (.r‘𝐽))
36	34, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → · = (.r‘𝐽))
37	36	eqcomd 2741	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (.r‘𝐽) = · )
38	37	oveqd 7422	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(.r‘𝐽)( 1 · 𝑏)) = (( 1 · 𝑎) · ( 1 · 𝑏)))
39	9, 15, 16, 7, 12, 17, 18	rngqiprnglinlem1 21252	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎) · ( 1 · 𝑏)) = ( 1 · (𝑎 · 𝑏)))
40	38, 39	eqtrd 2770	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(.r‘𝐽)( 1 · 𝑏)) = ( 1 · (𝑎 · 𝑏)))
41	33, 40	opeq12d 4857	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ⟨([𝑎] ∼ (.r‘𝑄)[𝑏] ∼ ), (( 1 · 𝑎)(.r‘𝐽)( 1 · 𝑏))⟩ = ⟨[(𝑎 · 𝑏)] ∼ , ( 1 · (𝑎 · 𝑏))⟩)
42	31, 41	eqtr2d 2771	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ⟨[(𝑎 · 𝑏)] ∼ , ( 1 · (𝑎 · 𝑏))⟩ = (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩(.r‘𝑃)⟨[𝑏] ∼ , ( 1 · 𝑏)⟩))
43	9	anim1i 615	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑅 ∈ Rng ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)))
44		3anass 1094	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ↔ (𝑅 ∈ Rng ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)))
45	43, 44	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵))
46	12, 17	rngcl 20124	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 · 𝑏) ∈ 𝐵)
47	45, 46	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
48		rngqiprngim.c	. . . . 5 ⊢ 𝐶 = (Base‘𝑄)
49		rngqiprngim.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)
50	9, 15, 16, 7, 12, 17, 18, 11, 4, 48, 1, 49	rngqiprngimfv 21259	. . . 4 ⊢ ((𝜑 ∧ (𝑎 · 𝑏) ∈ 𝐵) → (𝐹‘(𝑎 · 𝑏)) = ⟨[(𝑎 · 𝑏)] ∼ , ( 1 · (𝑎 · 𝑏))⟩)
51	47, 50	syldan 591	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎 · 𝑏)) = ⟨[(𝑎 · 𝑏)] ∼ , ( 1 · (𝑎 · 𝑏))⟩)
52	9, 15, 16, 7, 12, 17, 18, 11, 4, 48, 1, 49	rngqiprngimfv 21259	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
53	10, 52	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
54	9, 15, 16, 7, 12, 17, 18, 11, 4, 48, 1, 49	rngqiprngimfv 21259	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
55	21, 54	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
56	53, 55	oveq12d 7423	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝐹‘𝑎)(.r‘𝑃)(𝐹‘𝑏)) = (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩(.r‘𝑃)⟨[𝑏] ∼ , ( 1 · 𝑏)⟩))
57	42, 51, 56	3eqtr4d 2780	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(.r‘𝑃)(𝐹‘𝑏)))
58	57	ralrimivva 3187	1 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(.r‘𝑃)(𝐹‘𝑏)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  Vcvv 3459  ⟨cop 4607   ↦ cmpt 5201  ‘cfv 6531  (class class class)co 7405  [cec 8717  Basecbs 17228   ↾_s cress 17251  ._rcmulr 17272   /_s cqus 17519   ×_s cxps 17520   ~_QG cqg 19105  Rngcrng 20112  1_rcur 20141  Ringcrg 20193  2Idealc2idl 21210

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-ec 8721  df-qs 8725  df-map 8842  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-inf 9455  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-hom 17295  df-cco 17296  df-0g 17455  df-prds 17461  df-imas 17522  df-qus 17523  df-xps 17524  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-sbg 18921  df-subg 19106  df-eqg 19108  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-oppr 20297  df-subrng 20506  df-lss 20889  df-sra 21131  df-rgmod 21132  df-lidl 21169  df-2idl 21211

This theorem is referenced by:  rngqiprngho  21264