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Theorem rngqiprnglin 21640

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 2763	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
3		eqid 2763	. . . . 5 ⊢ (Base‘𝐽) = (Base‘𝐽)
4		rngqiprngim.q	. . . . . . 7 ⊢ 𝑄 = (𝑅 /_s ∼ )
5	4	ovexi 7483	. . . . . 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 19441	. . . . . 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 21625	. . . . . 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 19441	. . . . . 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 21625	. . . . . 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 21631	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (._r‘𝑄)[𝑏] ∼ ) ∈ (Base‘𝑄))
27		eqid 2763	. . . . . 6 ⊢ (._r‘𝐽) = (._r‘𝐽)
28	3, 27, 8, 20, 25	ringcld 20526	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(._r‘𝐽)( 1 · 𝑏)) ∈ (Base‘𝐽))
29		eqid 2763	. . . . 5 ⊢ (._r‘𝑄) = (._r‘𝑄)
30		eqid 2763	. . . . 5 ⊢ (._r‘𝑃) = (._r‘𝑃)
31	1, 2, 3, 6, 8, 14, 20, 23, 25, 26, 28, 29, 27, 30	xpsmul 17788	. . . 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 21630	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎 · 𝑏)] ∼ = ([𝑎] ∼ (._r‘𝑄)[𝑏] ∼ ))
33	32	eqcomd 2769	. . . . 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 17519	. . . . . . . . 9 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → · = (._r‘𝐽))
36	34, 35	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → · = (._r‘𝐽))
37	36	eqcomd 2769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (._r‘𝐽) = · )
38	37	oveqd 7466	. . . . . 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 21629	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎) · ( 1 · 𝑏)) = ( 1 · (𝑎 · 𝑏)))
40	38, 39	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(._r‘𝐽)( 1 · 𝑏)) = ( 1 · (𝑎 · 𝑏)))
41	33, 40	opeq12d 4851	. . . 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 2799	. . 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 20428	. . . . 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 21636	. . . 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 21636	. . . . 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 21636	. . . . 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 7467	. . 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 2808	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(._r‘𝑃)(𝐹‘𝑏)))
58	57	ralrimivva 3209	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 1570 ∈ wcel 2147 ∀wral 3080 Vcvv 3454 ⟨cop 4838 ↦ cmpt 5190 ‘cfv 6564 (class class class)co 7449 [cec 8751 Basecbs 17428 ↾_s cress 17449 ._rcmulr 17470 /_s cqus 17718 ×_s cxps 17719 ~_QG cqg 19373 Rngcrng 20416 1_rcur 20449 Ringcrg 20501 2Idealc2idl 21584

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-om 7910 df-1st 8033 df-2nd 8034 df-tpos 8279 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-1o 8512 df-2o 8513 df-er 8753 df-ec 8755 df-qs 8759 df-map 8885 df-ixp 8962 df-en 9010 df-dom 9011 df-sdom 9012 df-fin 9013 df-sup 9469 df-inf 9470 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-nn 12374 df-2 12443 df-3 12444 df-4 12445 df-5 12446 df-6 12447 df-7 12448 df-8 12449 df-9 12450 df-n0 12645 df-z 12732 df-dec 12853 df-uz 13004 df-fz 13678 df-struct 17366 df-sets 17383 df-slot 17401 df-ndx 17413 df-base 17429 df-ress 17450 df-plusg 17482 df-mulr 17483 df-sca 17485 df-vsca 17486 df-ip 17487 df-tset 17488 df-ple 17489 df-ds 17491 df-hom 17493 df-cco 17494 df-0g 17653 df-prds 17659 df-imas 17721 df-qus 17722 df-xps 17723 df-mgm 18857 df-sgrp 18949 df-mnd 18965 df-grp 19188 df-minusg 19189 df-sbg 19190 df-subg 19374 df-eqg 19376 df-cmn 20038 df-abl 20039 df-mgp 20403 df-rng 20417 df-ur 20450 df-ring 20503 df-oppr 20609 df-subrng 20840 df-lss 21249 df-sra 21490 df-rgmod 21491 df-lidl 21528 df-2idl 21585

This theorem is used by: rngqiprngho 21641

W3C validator