Theorem rngqiprngfu 21242

Description: The function value of 𝐹 at the constructed expected ring unity of 𝑅 is the ring unity of the product of the quotient with the two-sided ideal and the two-sided ideal. (Contributed by AV, 16-Mar-2025.)

Hypotheses

Ref	Expression
rngqiprngfu.r	⊢ (𝜑 → 𝑅 ∈ Rng)
rngqiprngfu.i	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
rngqiprngfu.j	⊢ 𝐽 = (𝑅 ↾s 𝐼)
rngqiprngfu.u	⊢ (𝜑 → 𝐽 ∈ Ring)
rngqiprngfu.b	⊢ 𝐵 = (Base‘𝑅)
rngqiprngfu.t	⊢ · = (.r‘𝑅)
rngqiprngfu.1	⊢ 1 = (1r‘𝐽)
rngqiprngfu.g	⊢ ∼ = (𝑅 ~QG 𝐼)
rngqiprngfu.q	⊢ 𝑄 = (𝑅 /s ∼ )
rngqiprngfu.v	⊢ (𝜑 → 𝑄 ∈ Ring)
rngqiprngfu.e	⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))
rngqiprngfu.m	⊢ − = (-g‘𝑅)
rngqiprngfu.a	⊢ + = (+g‘𝑅)
rngqiprngfu.n	⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )
rngqiprngfu.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)

Assertion

Ref	Expression
rngqiprngfu	⊢ (𝜑 → (𝐹‘𝑈) = ⟨[𝐸] ∼ , 1 ⟩)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐸   𝑥,𝑄   𝑥, ∼   𝜑,𝑥   𝑥,𝐼   𝑥,𝑈   𝑥, 1   𝑥, ·

Allowed substitution hints:   + (𝑥)   𝑅(𝑥)   𝐹(𝑥)   𝐽(𝑥)   − (𝑥)

Proof of Theorem rngqiprngfu

Step	Hyp	Ref	Expression
1		rngqiprngfu.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Rng)
2		rngqiprngfu.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
3		rngqiprngfu.j	. . . 4 ⊢ 𝐽 = (𝑅 ↾s 𝐼)
4		rngqiprngfu.u	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Ring)
5		rngqiprngfu.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
6		rngqiprngfu.t	. . . 4 ⊢ · = (.r‘𝑅)
7		rngqiprngfu.1	. . . 4 ⊢ 1 = (1r‘𝐽)
8		rngqiprngfu.g	. . . 4 ⊢ ∼ = (𝑅 ~QG 𝐼)
9		rngqiprngfu.q	. . . 4 ⊢ 𝑄 = (𝑅 /s ∼ )
10		rngqiprngfu.v	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Ring)
11		rngqiprngfu.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄))
12		rngqiprngfu.m	. . . 4 ⊢ − = (-g‘𝑅)
13		rngqiprngfu.a	. . . 4 ⊢ + = (+g‘𝑅)
14		rngqiprngfu.n	. . . 4 ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	rngqiprngfulem3 21238	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
16		eqid 2729	. . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄)
17		eqid 2729	. . . 4 ⊢ (𝑄 ×s 𝐽) = (𝑄 ×s 𝐽)
18		rngqiprngfu.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 18	rngqiprngimfv 21223	. . 3 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐵) → (𝐹‘𝑈) = ⟨[𝑈] ∼ , ( 1 · 𝑈)⟩)
20	15, 19	mpdan 687	. 2 ⊢ (𝜑 → (𝐹‘𝑈) = ⟨[𝑈] ∼ , ( 1 · 𝑈)⟩)
21	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	rngqiprngfulem4 21239	. . 3 ⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ )
22	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	rngqiprngfulem5 21240	. . 3 ⊢ (𝜑 → ( 1 · 𝑈) = 1 )
23	21, 22	opeq12d 4835	. 2 ⊢ (𝜑 → ⟨[𝑈] ∼ , ( 1 · 𝑈)⟩ = ⟨[𝐸] ∼ , 1 ⟩)
24	20, 23	eqtrd 2764	1 ⊢ (𝜑 → (𝐹‘𝑈) = ⟨[𝐸] ∼ , 1 ⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4585   ↦ cmpt 5176  ‘cfv 6486  (class class class)co 7353  [cec 8630  Basecbs 17138   ↾_s cress 17159  +_gcplusg 17179  ._rcmulr 17180   /_s cqus 17427   ×_s cxps 17428  -_gcsg 18832   ~_QG cqg 19019  Rngcrng 20055  1_rcur 20084  Ringcrg 20136  2Idealc2idl 21174

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-ec 8634  df-qs 8638  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-z 12490  df-dec 12610  df-uz 12754  df-fz 13429  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-0g 17363  df-imas 17430  df-qus 17431  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-grp 18833  df-minusg 18834  df-sbg 18835  df-subg 19020  df-nsg 19021  df-eqg 19022  df-cmn 19679  df-abl 19680  df-mgp 20044  df-rng 20056  df-ur 20085  df-ring 20138  df-oppr 20240  df-subrng 20449  df-lss 20853  df-sra 21095  df-rgmod 21096  df-lidl 21133  df-2idl 21175

This theorem is referenced by:  rngqiprngu  21243