Theorem rngqiprngfu 21356

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 21352	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
16		eqid 2752	. . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄)
17		eqid 2752	. . . 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 21337	. . 3 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐵) → (𝐹‘𝑈) = ⟨[𝑈] ∼ , ( 1 · 𝑈)⟩)
20	15, 19	mpdan 695	. 2 ⊢ (𝜑 → (𝐹‘𝑈) = ⟨[𝑈] ∼ , ( 1 · 𝑈)⟩)
21	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	rngqiprngfulem4 21353	. . 3 ⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ )
22	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	rngqiprngfulem5 21354	. . 3 ⊢ (𝜑 → ( 1 · 𝑈) = 1 )
23	21, 22	opeq12d 4829	. 2 ⊢ (𝜑 → ⟨[𝑈] ∼ , ( 1 · 𝑈)⟩ = ⟨[𝐸] ∼ , 1 ⟩)
24	20, 23	eqtrd 2787	1 ⊢ (𝜑 → (𝐹‘𝑈) = ⟨[𝐸] ∼ , 1 ⟩)

Syntax hints:   → wi 4   = wceq 1550   ∈ wcel 2132  ⟨cop 4578   ↦ cmpt 5171  ‘cfv 6506  (class class class)co 7381  [cec 8660  Basecbs 17217   ↾_s cress 17238  +_gcplusg 17258  ._rcmulr 17259   /_s cqus 17507   ×_s cxps 17508  -_gcsg 18949   ~_QG cqg 19136  Rngcrng 20170  1_rcur 20199  Ringcrg 20251  2Idealc2idl 21288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-tpos 8190  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-er 8662  df-ec 8664  df-qs 8668  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-sup 9374  df-inf 9375  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-nn 12197  df-2 12266  df-3 12267  df-4 12268  df-5 12269  df-6 12270  df-7 12271  df-8 12272  df-9 12273  df-n0 12468  df-z 12555  df-dec 12675  df-uz 12826  df-fz 13499  df-struct 17155  df-sets 17172  df-slot 17190  df-ndx 17202  df-base 17218  df-ress 17239  df-plusg 17271  df-mulr 17272  df-sca 17274  df-vsca 17275  df-ip 17276  df-tset 17277  df-ple 17278  df-ds 17280  df-0g 17442  df-imas 17510  df-qus 17511  df-mgm 18646  df-sgrp 18725  df-mnd 18741  df-grp 18950  df-minusg 18951  df-sbg 18952  df-subg 19137  df-nsg 19138  df-eqg 19139  df-cmn 19794  df-abl 19795  df-mgp 20159  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20354  df-subrng 20564  df-lss 20968  df-sra 21209  df-rgmod 21210  df-lidl 21247  df-2idl 21289

This theorem is referenced by:  rngqiprngu  21357