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Theorem rngqiprngghm 21188
Description: 𝐹 is a homomorphism of the additive groups of non-unital rings. (Contributed by AV, 24-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
rngqiprngghm (πœ‘ β†’ 𝐹 ∈ (𝑅 GrpHom 𝑃))
Distinct variable groups:   π‘₯,𝐢   π‘₯,𝐼   π‘₯,𝐡   πœ‘,π‘₯   π‘₯, ∼   π‘₯, 1   π‘₯, Β·   π‘₯,𝑅
Allowed substitution hints:   𝑃(π‘₯)   𝑄(π‘₯)   𝐹(π‘₯)   𝐽(π‘₯)
Proof of Theorem rngqiprngghm
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rng2idlring.b . 2 𝐡 = (Baseβ€˜π‘…)
2 eqid 2725 . 2 (Baseβ€˜π‘ƒ) = (Baseβ€˜π‘ƒ)
3 eqid 2725 . 2 (+gβ€˜π‘…) = (+gβ€˜π‘…)
4 eqid 2725 . 2 (+gβ€˜π‘ƒ) = (+gβ€˜π‘ƒ)
5 rng2idlring.r . . 3 (πœ‘ β†’ 𝑅 ∈ Rng)
6 rnggrp 20097 . . 3 (𝑅 ∈ Rng β†’ 𝑅 ∈ Grp)
75, 6syl 17 . 2 (πœ‘ β†’ 𝑅 ∈ Grp)
8 rng2idlring.i . . . 4 (πœ‘ β†’ 𝐼 ∈ (2Idealβ€˜π‘…))
9 rng2idlring.j . . . 4 𝐽 = (𝑅 β†Ύs 𝐼)
10 rng2idlring.u . . . 4 (πœ‘ β†’ 𝐽 ∈ Ring)
11 rng2idlring.t . . . 4 Β· = (.rβ€˜π‘…)
12 rng2idlring.1 . . . 4 1 = (1rβ€˜π½)
13 rngqiprngim.g . . . 4 ∼ = (𝑅 ~QG 𝐼)
14 rngqiprngim.q . . . 4 𝑄 = (𝑅 /s ∼ )
15 rngqiprngim.c . . . 4 𝐢 = (Baseβ€˜π‘„)
16 rngqiprngim.p . . . 4 𝑃 = (𝑄 Γ—s 𝐽)
175, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16rngqiprng 21185 . . 3 (πœ‘ β†’ 𝑃 ∈ Rng)
18 rnggrp 20097 . . 3 (𝑃 ∈ Rng β†’ 𝑃 ∈ Grp)
1917, 18syl 17 . 2 (πœ‘ β†’ 𝑃 ∈ Grp)
20 rngqiprngim.f . . . 4 𝐹 = (π‘₯ ∈ 𝐡 ↦ ⟨[π‘₯] ∼ , ( 1 Β· π‘₯)⟩)
215, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20rngqiprngimf 21186 . . 3 (πœ‘ β†’ 𝐹:𝐡⟢(𝐢 Γ— 𝐼))
225, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16rngqipbas 21184 . . . 4 (πœ‘ β†’ (Baseβ€˜π‘ƒ) = (𝐢 Γ— 𝐼))
2322feq3d 6704 . . 3 (πœ‘ β†’ (𝐹:𝐡⟢(Baseβ€˜π‘ƒ) ↔ 𝐹:𝐡⟢(𝐢 Γ— 𝐼)))
2421, 23mpbird 256 . 2 (πœ‘ β†’ 𝐹:𝐡⟢(Baseβ€˜π‘ƒ))
25 ringrng 20220 . . . . . . . . 9 (𝐽 ∈ Ring β†’ 𝐽 ∈ Rng)
2610, 25syl 17 . . . . . . . 8 (πœ‘ β†’ 𝐽 ∈ Rng)
279, 26eqeltrrid 2830 . . . . . . 7 (πœ‘ β†’ (𝑅 β†Ύs 𝐼) ∈ Rng)
285, 8, 27rng2idlnsg 21159 . . . . . 6 (πœ‘ β†’ 𝐼 ∈ (NrmSGrpβ€˜π‘…))
2928, 1, 13, 14ecqusaddd 19146 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ [(π‘Ž(+gβ€˜π‘…)𝑏)] ∼ = ([π‘Ž] ∼ (+gβ€˜π‘„)[𝑏] ∼ ))
305, 8, 9, 10, 1, 11, 12rngqiprngghmlem3 21178 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( 1 Β· (π‘Ž(+gβ€˜π‘…)𝑏)) = (( 1 Β· π‘Ž)(+gβ€˜π½)( 1 Β· 𝑏)))
3129, 30opeq12d 4878 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ⟨[(π‘Ž(+gβ€˜π‘…)𝑏)] ∼ , ( 1 Β· (π‘Ž(+gβ€˜π‘…)𝑏))⟩ = ⟨([π‘Ž] ∼ (+gβ€˜π‘„)[𝑏] ∼ ), (( 1 Β· π‘Ž)(+gβ€˜π½)( 1 Β· 𝑏))⟩)
32 eqid 2725 . . . . 5 (Baseβ€˜π‘„) = (Baseβ€˜π‘„)
33 eqid 2725 . . . . 5 (Baseβ€˜π½) = (Baseβ€˜π½)
3414ovexi 7447 . . . . . 6 𝑄 ∈ V
3534a1i 11 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑄 ∈ V)
3610adantr 479 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝐽 ∈ Ring)
37 simpl 481 . . . . . 6 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ π‘Ž ∈ 𝐡)
3813, 14, 1, 32quseccl0 19139 . . . . . 6 ((𝑅 ∈ Rng ∧ π‘Ž ∈ 𝐡) β†’ [π‘Ž] ∼ ∈ (Baseβ€˜π‘„))
395, 37, 38syl2an 594 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ [π‘Ž] ∼ ∈ (Baseβ€˜π‘„))
405, 8, 9, 10, 1, 11, 12rngqiprngghmlem1 21176 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ ( 1 Β· π‘Ž) ∈ (Baseβ€˜π½))
4140adantrr 715 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( 1 Β· π‘Ž) ∈ (Baseβ€˜π½))
42 simpr 483 . . . . . 6 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ 𝑏 ∈ 𝐡)
4313, 14, 1, 32quseccl0 19139 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑏 ∈ 𝐡) β†’ [𝑏] ∼ ∈ (Baseβ€˜π‘„))
445, 42, 43syl2an 594 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ [𝑏] ∼ ∈ (Baseβ€˜π‘„))
455, 8, 9, 10, 1, 11, 12rngqiprngghmlem1 21176 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ ( 1 Β· 𝑏) ∈ (Baseβ€˜π½))
4645adantrl 714 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( 1 Β· 𝑏) ∈ (Baseβ€˜π½))
4728, 1, 13, 14ecqusaddcl 19147 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ([π‘Ž] ∼ (+gβ€˜π‘„)[𝑏] ∼ ) ∈ (Baseβ€˜π‘„))
485, 8, 9, 10, 1, 11, 12rngqiprngghmlem2 21177 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (( 1 Β· π‘Ž)(+gβ€˜π½)( 1 Β· 𝑏)) ∈ (Baseβ€˜π½))
49 eqid 2725 . . . . 5 (+gβ€˜π‘„) = (+gβ€˜π‘„)
50 eqid 2725 . . . . 5 (+gβ€˜π½) = (+gβ€˜π½)
5116, 32, 33, 35, 36, 39, 41, 44, 46, 47, 48, 49, 50, 4xpsadd 17550 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩(+gβ€˜π‘ƒ)⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩) = ⟨([π‘Ž] ∼ (+gβ€˜π‘„)[𝑏] ∼ ), (( 1 Β· π‘Ž)(+gβ€˜π½)( 1 Β· 𝑏))⟩)
5231, 51eqtr4d 2768 . . 3 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ⟨[(π‘Ž(+gβ€˜π‘…)𝑏)] ∼ , ( 1 Β· (π‘Ž(+gβ€˜π‘…)𝑏))⟩ = (⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩(+gβ€˜π‘ƒ)⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩))
535adantr 479 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑅 ∈ Rng)
5437adantl 480 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ 𝐡)
5542adantl 480 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ 𝐡)
561, 3rngacl 20101 . . . . 5 ((𝑅 ∈ Rng ∧ π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (π‘Ž(+gβ€˜π‘…)𝑏) ∈ 𝐡)
5753, 54, 55, 56syl3anc 1368 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(+gβ€˜π‘…)𝑏) ∈ 𝐡)
585, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20rngqiprngimfv 21187 . . . 4 ((πœ‘ ∧ (π‘Ž(+gβ€˜π‘…)𝑏) ∈ 𝐡) β†’ (πΉβ€˜(π‘Ž(+gβ€˜π‘…)𝑏)) = ⟨[(π‘Ž(+gβ€˜π‘…)𝑏)] ∼ , ( 1 Β· (π‘Ž(+gβ€˜π‘…)𝑏))⟩)
5957, 58syldan 589 . . 3 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜(π‘Ž(+gβ€˜π‘…)𝑏)) = ⟨[(π‘Ž(+gβ€˜π‘…)𝑏)] ∼ , ( 1 Β· (π‘Ž(+gβ€˜π‘…)𝑏))⟩)
605, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20rngqiprngimfv 21187 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (πΉβ€˜π‘Ž) = ⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩)
6160adantrr 715 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜π‘Ž) = ⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩)
625, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20rngqiprngimfv 21187 . . . . 5 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (πΉβ€˜π‘) = ⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩)
6362adantrl 714 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜π‘) = ⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩)
6461, 63oveq12d 7431 . . 3 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((πΉβ€˜π‘Ž)(+gβ€˜π‘ƒ)(πΉβ€˜π‘)) = (⟨[π‘Ž] ∼ , ( 1 Β· π‘Ž)⟩(+gβ€˜π‘ƒ)⟨[𝑏] ∼ , ( 1 Β· 𝑏)⟩))
6552, 59, 643eqtr4d 2775 . 2 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜(π‘Ž(+gβ€˜π‘…)𝑏)) = ((πΉβ€˜π‘Ž)(+gβ€˜π‘ƒ)(πΉβ€˜π‘)))
661, 2, 3, 4, 7, 19, 24, 65isghmd 19178 1 (πœ‘ β†’ 𝐹 ∈ (𝑅 GrpHom 𝑃))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463  βŸ¨cop 4631   ↦ cmpt 5227   Γ— cxp 5671  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7413  [cec 8716  Basecbs 17174   β†Ύs cress 17203  +gcplusg 17227  .rcmulr 17228   /s cqus 17481   Γ—s cxps 17482  Grpcgrp 18889   ~QG cqg 19076   GrpHom cghm 19166  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-nsg 19078  df-eqg 19079  df-ghm 19167  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:  rngqiprngimf1  21189  rngqiprngho  21192
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