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Theorem rngqiprnglin 21181
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 2727 . . . . 5 (Baseβ€˜π‘„) = (Baseβ€˜π‘„)
3 eqid 2727 . . . . 5 (Baseβ€˜π½) = (Baseβ€˜π½)
4 rngqiprngim.q . . . . . . 7 𝑄 = (𝑅 /s ∼ )
54ovexi 7448 . . . . . 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 480 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝐽 ∈ Ring)
9 rng2idlring.r . . . . . 6 (πœ‘ β†’ 𝑅 ∈ Rng)
10 simpl 482 . . . . . 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 19131 . . . . . 6 ((𝑅 ∈ Rng ∧ π‘Ž ∈ 𝐡) β†’ [π‘Ž] ∼ ∈ (Baseβ€˜π‘„))
149, 10, 13syl2an 595 . . . . 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 21166 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ ( 1 Β· π‘Ž) ∈ (Baseβ€˜π½))
2010, 19sylan2 592 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( 1 Β· π‘Ž) ∈ (Baseβ€˜π½))
21 simpr 484 . . . . . 6 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ 𝑏 ∈ 𝐡)
2211, 4, 12, 2quseccl0 19131 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑏 ∈ 𝐡) β†’ [𝑏] ∼ ∈ (Baseβ€˜π‘„))
239, 21, 22syl2an 595 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ [𝑏] ∼ ∈ (Baseβ€˜π‘„))
249, 15, 16, 7, 12, 17, 18rngqiprngghmlem1 21166 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ ( 1 Β· 𝑏) ∈ (Baseβ€˜π½))
2521, 24sylan2 592 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( 1 Β· 𝑏) ∈ (Baseβ€˜π½))
269, 15, 16, 7, 12, 17, 18, 11, 4rngqiprnglinlem3 21172 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ([π‘Ž] ∼ (.rβ€˜π‘„)[𝑏] ∼ ) ∈ (Baseβ€˜π‘„))
27 eqid 2727 . . . . . 6 (.rβ€˜π½) = (.rβ€˜π½)
283, 27, 8, 20, 25ringcld 20188 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (( 1 Β· π‘Ž)(.rβ€˜π½)( 1 Β· 𝑏)) ∈ (Baseβ€˜π½))
29 eqid 2727 . . . . 5 (.rβ€˜π‘„) = (.rβ€˜π‘„)
30 eqid 2727 . . . . 5 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
311, 2, 3, 6, 8, 14, 20, 23, 25, 26, 28, 29, 27, 30xpsmul 17548 . . . 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 21171 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ [(π‘Ž Β· 𝑏)] ∼ = ([π‘Ž] ∼ (.rβ€˜π‘„)[𝑏] ∼ ))
3332eqcomd 2733 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ([π‘Ž] ∼ (.rβ€˜π‘„)[𝑏] ∼ ) = [(π‘Ž Β· 𝑏)] ∼ )
3415adantr 480 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝐼 ∈ (2Idealβ€˜π‘…))
3516, 17ressmulr 17279 . . . . . . . . 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 2733 . . . . . . 7 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (.rβ€˜π½) = Β· )
3837oveqd 7431 . . . . . 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 21170 . . . . . 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 2767 . . . . 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 4877 . . . 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 2768 . . 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 614 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (𝑅 ∈ Rng ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)))
44 3anass 1093 . . . . . 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 20095 . . . . 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 21177 . . . 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 590 . . 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 21177 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (πΉβ€˜π‘Ž) = ⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩)
5310, 52sylan2 592 . . . 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 21177 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (πΉβ€˜π‘) = ⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩)
5521, 54sylan2 592 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜π‘) = ⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩)
5653, 55oveq12d 7432 . . 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 2777 . 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 3195 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  Vcvv 3469  βŸ¨cop 4630   ↦ cmpt 5225  β€˜cfv 6542  (class class class)co 7414  [cec 8716  Basecbs 17171   β†Ύs cress 17200  .rcmulr 17225   /s cqus 17478   Γ—s cxps 17479   ~QG cqg 19068  Rngcrng 20083  1rcur 20112  Ringcrg 20164  2Idealc2idl 21132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  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 8838  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-mulr 17238  df-sca 17240  df-vsca 17241  df-ip 17242  df-tset 17243  df-ple 17244  df-ds 17246  df-hom 17248  df-cco 17249  df-0g 17414  df-prds 17420  df-imas 17481  df-qus 17482  df-xps 17483  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-grp 18884  df-minusg 18885  df-sbg 18886  df-subg 19069  df-eqg 19071  df-cmn 19728  df-abl 19729  df-mgp 20066  df-rng 20084  df-ur 20113  df-ring 20166  df-oppr 20262  df-subrng 20472  df-lss 20805  df-sra 21047  df-rgmod 21048  df-lidl 21093  df-2idl 21133
This theorem is referenced by:  rngqiprngho  21182
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