Theorem rngqiprnglin 21330

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 7465	. . . . . 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 19216	. . . . . 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 21315	. . . . . 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 19216	. . . . . 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 21315	. . . . . 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 21321	. . . . 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 20277	. . . . 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 17622	. . . 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 21320	. . . . . 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 17353	. . . . . . . . 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 7448	. . . . . 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 21319	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎) · ( 1 · 𝑏)) = ( 1 · (𝑎 · 𝑏)))
40	38, 39	eqtrd 2775	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(.r‘𝐽)( 1 · 𝑏)) = ( 1 · (𝑎 · 𝑏)))
41	33, 40	opeq12d 4886	. . . 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 2776	. . 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 20182	. . . . 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 21326	. . . 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 21326	. . . . 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 21326	. . . . 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 7449	. . 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 2785	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(.r‘𝑃)(𝐹‘𝑏)))
58	57	ralrimivva 3200	1 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(.r‘𝑃)(𝐹‘𝑏)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478  ⟨cop 4637   ↦ cmpt 5231  ‘cfv 6563  (class class class)co 7431  [cec 8742  Basecbs 17245   ↾_s cress 17274  ._rcmulr 17299   /_s cqus 17552   ×_s cxps 17553   ~_QG cqg 19153  Rngcrng 20170  1_rcur 20199  Ringcrg 20251  2Idealc2idl 21277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-prds 17494  df-imas 17555  df-qus 17556  df-xps 17557  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-eqg 19156  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-subrng 20563  df-lss 20948  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-2idl 21278

This theorem is referenced by:  rngqiprngho  21331