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Theorem rngqiprnglin 21604

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 = (1_r‘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 2760	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
3		eqid 2760	. . . . 5 ⊢ (Base‘𝐽) = (Base‘𝐽)
4		rngqiprngim.q	. . . . . . 7 ⊢ 𝑄 = (𝑅 /_s ∼ )
5	4	ovexi 7455	. . . . . 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 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝐽 ∈ Ring)
9		rng2idlring.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Rng)
10		simpl 488	. . . . . 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 19406	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵) → [𝑎] ∼ ∈ (Base‘𝑄))
14	9, 10, 13	syl2an 608	. . . . 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 = (1_r‘𝐽)
19	9, 15, 16, 7, 12, 17, 18	rngqiprngghmlem1 21589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ( 1 · 𝑎) ∈ (Base‘𝐽))
20	10, 19	sylan2 605	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑎) ∈ (Base‘𝐽))
21		simpr 490	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
22	11, 4, 12, 2	quseccl0 19406	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑏 ∈ 𝐵) → [𝑏] ∼ ∈ (Base‘𝑄))
23	9, 21, 22	syl2an 608	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑏] ∼ ∈ (Base‘𝑄))
24	9, 15, 16, 7, 12, 17, 18	rngqiprngghmlem1 21589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ( 1 · 𝑏) ∈ (Base‘𝐽))
25	21, 24	sylan2 605	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑏) ∈ (Base‘𝐽))
26	9, 15, 16, 7, 12, 17, 18, 11, 4	rngqiprnglinlem3 21595	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (._r‘𝑄)[𝑏] ∼ ) ∈ (Base‘𝑄))
27		eqid 2760	. . . . . 6 ⊢ (._r‘𝐽) = (._r‘𝐽)
28	3, 27, 8, 20, 25	ringcld 20490	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(._r‘𝐽)( 1 · 𝑏)) ∈ (Base‘𝐽))
29		eqid 2760	. . . . 5 ⊢ (._r‘𝑄) = (._r‘𝑄)
30		eqid 2760	. . . . 5 ⊢ (._r‘𝑃) = (._r‘𝑃)
31	1, 2, 3, 6, 8, 14, 20, 23, 25, 26, 28, 29, 27, 30	xpsmul 17753	. . . 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 21594	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎 · 𝑏)] ∼ = ([𝑎] ∼ (._r‘𝑄)[𝑏] ∼ ))
33	32	eqcomd 2766	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (._r‘𝑄)[𝑏] ∼ ) = [(𝑎 · 𝑏)] ∼ )
34	15	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝐼 ∈ (2Ideal‘𝑅))
35	16, 17	ressmulr 17484	. . . . . . . . 9 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → · = (._r‘𝐽))
36	34, 35	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → · = (._r‘𝐽))
37	36	eqcomd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (._r‘𝐽) = · )
38	37	oveqd 7438	. . . . . 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 21593	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎) · ( 1 · 𝑏)) = ( 1 · (𝑎 · 𝑏)))
40	38, 39	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(._r‘𝐽)( 1 · 𝑏)) = ( 1 · (𝑎 · 𝑏)))
41	33, 40	opeq12d 4847	. . . 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 2796	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ⟨[(𝑎 · 𝑏)] ∼ , ( 1 · (𝑎 · 𝑏))⟩ = (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩(._r‘𝑃)⟨[𝑏] ∼ , ( 1 · 𝑏)⟩))
43	9	anim1i 627	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑅 ∈ Rng ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)))
44		3anass 1111	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ↔ (𝑅 ∈ Rng ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)))
45	43, 44	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵))
46	12, 17	rngcl 20392	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 · 𝑏) ∈ 𝐵)
47	45, 46	syl 18	. . . 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 21600	. . . 4 ⊢ ((𝜑 ∧ (𝑎 · 𝑏) ∈ 𝐵) → (𝐹‘(𝑎 · 𝑏)) = ⟨[(𝑎 · 𝑏)] ∼ , ( 1 · (𝑎 · 𝑏))⟩)
51	47, 50	syldan 603	. . 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 21600	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
53	10, 52	sylan2 605	. . . 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 21600	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
55	21, 54	sylan2 605	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
56	53, 55	oveq12d 7439	. . 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 2805	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(._r‘𝑃)(𝐹‘𝑏)))
58	57	ralrimivva 3205	1 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(._r‘𝑃)(𝐹‘𝑏)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 Vcvv 3450 ⟨cop 4834 ↦ cmpt 5185 ‘cfv 6539 (class class class)co 7421 [cec 8719 Basecbs 17393 ↾_s cress 17414 ._rcmulr 17435 /_s cqus 17683 ×_s cxps 17684 ~_QG cqg 19338 Rngcrng 20380 1_rcur 20413 Ringcrg 20465 2Idealc2idl 21548

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-tpos 8247 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-ec 8723 df-qs 8727 df-map 8853 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-sup 9438 df-inf 9439 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-fz 13644 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-hom 17458 df-cco 17459 df-0g 17618 df-prds 17624 df-imas 17686 df-qus 17687 df-xps 17688 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-grp 19153 df-minusg 19154 df-sbg 19155 df-subg 19339 df-eqg 19341 df-cmn 20002 df-abl 20003 df-mgp 20367 df-rng 20381 df-ur 20414 df-ring 20467 df-oppr 20573 df-subrng 20804 df-lss 21213 df-sra 21454 df-rgmod 21455 df-lidl 21492 df-2idl 21549

This theorem is used by: rngqiprngho 21605

W3C validator