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Theorem rngqiprnglin 21191
Description: 𝐹 is linear with respect to the multiplication. (Contributed by AV, 28-Feb-2025.)
Hypotheses
Ref Expression
rng2idlring.r (πœ‘ β†’ 𝑅 ∈ Rng)
rng2idlring.i (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))
rng2idlring.j 𝐽 = (𝑅 β†Ύs 𝐼)
rng2idlring.u (πœ‘ β†’ 𝐽 ∈ Ring)
rng2idlring.b 𝐡 = (Baseβ€˜π‘…)
rng2idlring.t Β· = (.rβ€˜π‘…)
rng2idlring.1 1 = (1rβ€˜π½)
rngqiprngim.g ∼ = (𝑅 ~QG 𝐼)
rngqiprngim.q 𝑄 = (𝑅 /s ∼ )
rngqiprngim.c 𝐢 = (Baseβ€˜π‘„)
rngqiprngim.p 𝑃 = (𝑄 Γ—s 𝐽)
rngqiprngim.f 𝐹 = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)
Assertion
Ref Expression
rngqiprnglin (πœ‘ β†’ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (πΉβ€˜(π‘Ž Β· 𝑏)) = ((πΉβ€˜π‘Ž)(.rβ€˜π‘ƒ)(πΉβ€˜π‘)))
Distinct variable groups:   π‘₯,𝐢   π‘₯,𝐼   π‘₯,𝐡   πœ‘,π‘₯   π‘₯, ∼   π‘₯, 1   π‘₯, Β·   𝐡,π‘Ž,𝑏   𝐹,π‘Ž,𝑏   𝑃,π‘Ž,𝑏   𝑅,π‘Ž,𝑏,π‘₯   πœ‘,π‘Ž,𝑏   𝐽,π‘Ž   𝑄,π‘Ž   𝐢,π‘Ž,𝑏   𝐼,π‘Ž,𝑏   ∼ ,π‘Ž   1 ,π‘Ž   Β· ,π‘Ž
Allowed substitution hints:   𝑃(π‘₯)   𝑄(π‘₯,𝑏)   ∼ (𝑏)   Β· (𝑏)   1 (𝑏)   𝐹(π‘₯)   𝐽(π‘₯,𝑏)
Proof of Theorem rngqiprnglin
StepHypRef Expression
1 rngqiprngim.p . . . . 5 𝑃 = (𝑄 Γ—s 𝐽)
2 eqid 2725 . . . . 5 (Baseβ€˜π‘„) = (Baseβ€˜π‘„)
3 eqid 2725 . . . . 5 (Baseβ€˜π½) = (Baseβ€˜π½)
4 rngqiprngim.q . . . . . . 7 𝑄 = (𝑅 /s ∼ )
54ovexi 7447 . . . . . 6 𝑄 ∈ V
65a1i 11 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑄 ∈ V)
7 rng2idlring.u . . . . . 6 (πœ‘ β†’ 𝐽 ∈ Ring)
87adantr 479 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝐽 ∈ Ring)
9 rng2idlring.r . . . . . 6 (πœ‘ β†’ 𝑅 ∈ Rng)
10 simpl 481 . . . . . 6 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ π‘Ž ∈ 𝐡)
11 rngqiprngim.g . . . . . . 7 ∼ = (𝑅 ~QG 𝐼)
12 rng2idlring.b . . . . . . 7 𝐡 = (Baseβ€˜π‘…)
1311, 4, 12, 2quseccl0 19139 . . . . . 6 ((𝑅 ∈ Rng ∧ π‘Ž ∈ 𝐡) β†’ [π‘Ž] ∼ ∈ (Baseβ€˜π‘„))
149, 10, 13syl2an 594 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ [π‘Ž] ∼ ∈ (Baseβ€˜π‘„))
15 rng2idlring.i . . . . . . 7 (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))
16 rng2idlring.j . . . . . . 7 𝐽 = (𝑅 β†Ύs 𝐼)
17 rng2idlring.t . . . . . . 7 Β· = (.rβ€˜π‘…)
18 rng2idlring.1 . . . . . . 7 1 = (1rβ€˜π½)
199, 15, 16, 7, 12, 17, 18rngqiprngghmlem1 21176 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ ( 1 Β· π‘Ž) ∈ (Baseβ€˜π½))
2010, 19sylan2 591 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( 1 Β· π‘Ž) ∈ (Baseβ€˜π½))
21 simpr 483 . . . . . 6 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ 𝑏 ∈ 𝐡)
2211, 4, 12, 2quseccl0 19139 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑏 ∈ 𝐡) β†’ [𝑏] ∼ ∈ (Baseβ€˜π‘„))
239, 21, 22syl2an 594 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ [𝑏] ∼ ∈ (Baseβ€˜π‘„))
249, 15, 16, 7, 12, 17, 18rngqiprngghmlem1 21176 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ ( 1 Β· 𝑏) ∈ (Baseβ€˜π½))
2521, 24sylan2 591 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( 1 Β· 𝑏) ∈ (Baseβ€˜π½))
269, 15, 16, 7, 12, 17, 18, 11, 4rngqiprnglinlem3 21182 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ([π‘Ž] ∼ (.rβ€˜π‘„)[𝑏] ∼ ) ∈ (Baseβ€˜π‘„))
27 eqid 2725 . . . . . 6 (.rβ€˜π½) = (.rβ€˜π½)
283, 27, 8, 20, 25ringcld 20198 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (( 1 Β· π‘Ž)(.rβ€˜π½)( 1 Β· 𝑏)) ∈ (Baseβ€˜π½))
29 eqid 2725 . . . . 5 (.rβ€˜π‘„) = (.rβ€˜π‘„)
30 eqid 2725 . . . . 5 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
311, 2, 3, 6, 8, 14, 20, 23, 25, 26, 28, 29, 27, 30xpsmul 17551 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩(.rβ€˜π‘ƒ)⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩) = ⟨([π‘Ž] ∼ (.rβ€˜π‘„)[𝑏] ∼ ), (( 1 Β· π‘Ž)(.rβ€˜π½)( 1 Β· 𝑏))⟩)
329, 15, 16, 7, 12, 17, 18, 11, 4rngqiprnglinlem2 21181 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ [(π‘Ž Β· 𝑏)] ∼ = ([π‘Ž] ∼ (.rβ€˜π‘„)[𝑏] ∼ ))
3332eqcomd 2731 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ([π‘Ž] ∼ (.rβ€˜π‘„)[𝑏] ∼ ) = [(π‘Ž Β· 𝑏)] ∼ )
3415adantr 479 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝐼 ∈ (2Idealβ€˜π‘…))
3516, 17ressmulr 17282 . . . . . . . . 9 (𝐼 ∈ (2Idealβ€˜π‘…) β†’ Β· = (.rβ€˜π½))
3634, 35syl 17 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ Β· = (.rβ€˜π½))
3736eqcomd 2731 . . . . . . 7 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (.rβ€˜π½) = Β· )
3837oveqd 7430 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (( 1 Β· π‘Ž)(.rβ€˜π½)( 1 Β· 𝑏)) = (( 1 Β· π‘Ž) Β· ( 1 Β· 𝑏)))
399, 15, 16, 7, 12, 17, 18rngqiprnglinlem1 21180 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (( 1 Β· π‘Ž) Β· ( 1 Β· 𝑏)) = ( 1 Β· (π‘Ž Β· 𝑏)))
4038, 39eqtrd 2765 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (( 1 Β· π‘Ž)(.rβ€˜π½)( 1 Β· 𝑏)) = ( 1 Β· (π‘Ž Β· 𝑏)))
4133, 40opeq12d 4878 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ⟨([π‘Ž] ∼ (.rβ€˜π‘„)[𝑏] ∼ ), (( 1 Β· π‘Ž)(.rβ€˜π½)( 1 Β· 𝑏))⟩ = ⟨[(π‘Ž Β· 𝑏)] ∼ , ( 1 Β· (π‘Ž Β· 𝑏))⟩)
4231, 41eqtr2d 2766 . . 3 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ⟨[(π‘Ž Β· 𝑏)] ∼ , ( 1 Β· (π‘Ž Β· 𝑏))⟩ = (⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩(.rβ€˜π‘ƒ)⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩))
439anim1i 613 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (𝑅 ∈ Rng ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)))
44 3anass 1092 . . . . . 6 ((𝑅 ∈ Rng ∧ π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) ↔ (𝑅 ∈ Rng ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)))
4543, 44sylibr 233 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (𝑅 ∈ Rng ∧ π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡))
4612, 17rngcl 20103 . . . . 5 ((𝑅 ∈ Rng ∧ π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (π‘Ž Β· 𝑏) ∈ 𝐡)
4745, 46syl 17 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž Β· 𝑏) ∈ 𝐡)
48 rngqiprngim.c . . . . 5 𝐢 = (Baseβ€˜π‘„)
49 rngqiprngim.f . . . . 5 𝐹 = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)
509, 15, 16, 7, 12, 17, 18, 11, 4, 48, 1, 49rngqiprngimfv 21187 . . . 4 ((πœ‘ ∧ (π‘Ž Β· 𝑏) ∈ 𝐡) β†’ (πΉβ€˜(π‘Ž Β· 𝑏)) = ⟨[(π‘Ž Β· 𝑏)] ∼ , ( 1 Β· (π‘Ž Β· 𝑏))⟩)
5147, 50syldan 589 . . 3 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜(π‘Ž Β· 𝑏)) = ⟨[(π‘Ž Β· 𝑏)] ∼ , ( 1 Β· (π‘Ž Β· 𝑏))⟩)
529, 15, 16, 7, 12, 17, 18, 11, 4, 48, 1, 49rngqiprngimfv 21187 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (πΉβ€˜π‘Ž) = ⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩)
5310, 52sylan2 591 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜π‘Ž) = ⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩)
549, 15, 16, 7, 12, 17, 18, 11, 4, 48, 1, 49rngqiprngimfv 21187 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (πΉβ€˜π‘) = ⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩)
5521, 54sylan2 591 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜π‘) = ⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩)
5653, 55oveq12d 7431 . . 3 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((πΉβ€˜π‘Ž)(.rβ€˜π‘ƒ)(πΉβ€˜π‘)) = (⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩(.rβ€˜π‘ƒ)⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩))
5742, 51, 563eqtr4d 2775 . 2 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜(π‘Ž Β· 𝑏)) = ((πΉβ€˜π‘Ž)(.rβ€˜π‘ƒ)(πΉβ€˜π‘)))
5857ralrimivva 3191 1 (πœ‘ β†’ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (πΉβ€˜(π‘Ž Β· 𝑏)) = ((πΉβ€˜π‘Ž)(.rβ€˜π‘ƒ)(πΉβ€˜π‘)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  Vcvv 3463  βŸ¨cop 4631   ↦ cmpt 5227  β€˜cfv 6543  (class class class)co 7413  [cec 8716  Basecbs 17174   β†Ύs cress 17203  .rcmulr 17228   /s cqus 17481   Γ—s cxps 17482   ~QG cqg 19076  Rngcrng 20091  1rcur 20120  Ringcrg 20172  2Idealc2idl 21142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-ec 8720  df-qs 8724  df-map 8840  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9460  df-inf 9461  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17417  df-prds 17423  df-imas 17484  df-qus 17485  df-xps 17486  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-grp 18892  df-minusg 18893  df-sbg 18894  df-subg 19077  df-eqg 19079  df-cmn 19736  df-abl 19737  df-mgp 20074  df-rng 20092  df-ur 20121  df-ring 20174  df-oppr 20272  df-subrng 20482  df-lss 20815  df-sra 21057  df-rgmod 21058  df-lidl 21103  df-2idl 21143
This theorem is referenced by:  rngqiprngho  21192
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