Theorem rngqiprngfu 21284

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 21280	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
16		eqid 2737	. . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄)
17		eqid 2737	. . . 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 21265	. . 3 ⊢ ((𝜑 ∧ 𝑈 ∈ 𝐵) → (𝐹‘𝑈) = ⟨[𝑈] ∼ , ( 1 · 𝑈)⟩)
20	15, 19	mpdan 688	. 2 ⊢ (𝜑 → (𝐹‘𝑈) = ⟨[𝑈] ∼ , ( 1 · 𝑈)⟩)
21	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	rngqiprngfulem4 21281	. . 3 ⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ )
22	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	rngqiprngfulem5 21282	. . 3 ⊢ (𝜑 → ( 1 · 𝑈) = 1 )
23	21, 22	opeq12d 4839	. 2 ⊢ (𝜑 → ⟨[𝑈] ∼ , ( 1 · 𝑈)⟩ = ⟨[𝐸] ∼ , 1 ⟩)
24	20, 23	eqtrd 2772	1 ⊢ (𝜑 → (𝐹‘𝑈) = ⟨[𝐸] ∼ , 1 ⟩)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   ↦ cmpt 5181  ‘cfv 6500  (class class class)co 7368  [cec 8643  Basecbs 17148   ↾_s cress 17169  +_gcplusg 17189  ._rcmulr 17190   /_s cqus 17438   ×_s cxps 17439  -_gcsg 18877   ~_QG cqg 19064  Rngcrng 20099  1_rcur 20128  Ringcrg 20180  2Idealc2idl 21216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-tpos 8178  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-ec 8647  df-qs 8651  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-sca 17205  df-vsca 17206  df-ip 17207  df-tset 17208  df-ple 17209  df-ds 17211  df-0g 17373  df-imas 17441  df-qus 17442  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-minusg 18879  df-sbg 18880  df-subg 19065  df-nsg 19066  df-eqg 19067  df-cmn 19723  df-abl 19724  df-mgp 20088  df-rng 20100  df-ur 20129  df-ring 20182  df-oppr 20285  df-subrng 20491  df-lss 20895  df-sra 21137  df-rgmod 21138  df-lidl 21175  df-2idl 21217

This theorem is referenced by:  rngqiprngu  21285