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Theorem rngqiprnglin 21599

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 2761	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
3		eqid 2761	. . . . 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 19401	. . . . . 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 21584	. . . . . 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 19401	. . . . . 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 21584	. . . . . 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 21590	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (._r‘𝑄)[𝑏] ∼ ) ∈ (Base‘𝑄))
27		eqid 2761	. . . . . 6 ⊢ (._r‘𝐽) = (._r‘𝐽)
28	3, 27, 8, 20, 25	ringcld 20485	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(._r‘𝐽)( 1 · 𝑏)) ∈ (Base‘𝐽))
29		eqid 2761	. . . . 5 ⊢ (._r‘𝑄) = (._r‘𝑄)
30		eqid 2761	. . . . 5 ⊢ (._r‘𝑃) = (._r‘𝑃)
31	1, 2, 3, 6, 8, 14, 20, 23, 25, 26, 28, 29, 27, 30	xpsmul 17748	. . . 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 21589	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎 · 𝑏)] ∼ = ([𝑎] ∼ (._r‘𝑄)[𝑏] ∼ ))
33	32	eqcomd 2767	. . . . 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 17479	. . . . . . . . 9 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → · = (._r‘𝐽))
36	34, 35	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → · = (._r‘𝐽))
37	36	eqcomd 2767	. . . . . . 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 21588	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎) · ( 1 · 𝑏)) = ( 1 · (𝑎 · 𝑏)))
40	38, 39	eqtrd 2796	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(._r‘𝐽)( 1 · 𝑏)) = ( 1 · (𝑎 · 𝑏)))
41	33, 40	opeq12d 4841	. . . 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 2797	. . 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 20387	. . . . 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 21595	. . . 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 21595	. . . . 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 21595	. . . . 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 2806	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(._r‘𝑃)(𝐹‘𝑏)))
58	57	ralrimivva 3206	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 2145 ∀wral 3077 Vcvv 3451 ⟨cop 4590 ↦ cmpt 5186 ‘cfv 6539 (class class class)co 7421 [cec 8716 Basecbs 17388 ↾_s cress 17409 ._rcmulr 17430 /_s cqus 17678 ×_s cxps 17679 ~_QG cqg 19333 Rngcrng 20375 1_rcur 20408 Ringcrg 20460 2Idealc2idl 21543

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7752 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3900 df-un 3902 df-in 3904 df-ss 3915 df-pss 4035 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 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 8244 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-2o 8478 df-er 8718 df-ec 8720 df-qs 8724 df-map 8850 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9435 df-inf 9436 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-7 12411 df-8 12412 df-9 12413 df-n0 12608 df-z 12695 df-dec 12816 df-uz 12967 df-fz 13641 df-struct 17326 df-sets 17343 df-slot 17361 df-ndx 17373 df-base 17389 df-ress 17410 df-plusg 17442 df-mulr 17443 df-sca 17445 df-vsca 17446 df-ip 17447 df-tset 17448 df-ple 17449 df-ds 17451 df-hom 17453 df-cco 17454 df-0g 17613 df-prds 17619 df-imas 17681 df-qus 17682 df-xps 17683 df-mgm 18817 df-sgrp 18909 df-mnd 18925 df-grp 19148 df-minusg 19149 df-sbg 19150 df-subg 19334 df-eqg 19336 df-cmn 19997 df-abl 19998 df-mgp 20362 df-rng 20376 df-ur 20409 df-ring 20462 df-oppr 20568 df-subrng 20799 df-lss 21208 df-sra 21449 df-rgmod 21450 df-lidl 21487 df-2idl 21544

This theorem is used by: rngqiprngho 21600

W3C validator