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Theorem rngqiprngghm 21327
Description: 𝐹 is a homomorphism of the additive groups of non-unital rings. (Contributed by AV, 24-Feb-2025.)
Hypotheses
Ref Expression
rng2idlring.r (𝜑𝑅 ∈ Rng)
rng2idlring.i (𝜑𝐼 ∈ (2Ideal‘𝑅))
rng2idlring.j 𝐽 = (𝑅s 𝐼)
rng2idlring.u (𝜑𝐽 ∈ Ring)
rng2idlring.b 𝐵 = (Base‘𝑅)
rng2idlring.t · = (.r𝑅)
rng2idlring.1 1 = (1r𝐽)
rngqiprngim.g = (𝑅 ~QG 𝐼)
rngqiprngim.q 𝑄 = (𝑅 /s )
rngqiprngim.c 𝐶 = (Base‘𝑄)
rngqiprngim.p 𝑃 = (𝑄 ×s 𝐽)
rngqiprngim.f 𝐹 = (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)
Assertion
Ref Expression
rngqiprngghm (𝜑𝐹 ∈ (𝑅 GrpHom 𝑃))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐼   𝑥,𝐵   𝜑,𝑥   𝑥,   𝑥, 1   𝑥, ·   𝑥,𝑅
Allowed substitution hints:   𝑃(𝑥)   𝑄(𝑥)   𝐹(𝑥)   𝐽(𝑥)

Proof of Theorem rngqiprngghm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rng2idlring.b . 2 𝐵 = (Base‘𝑅)
2 eqid 2735 . 2 (Base‘𝑃) = (Base‘𝑃)
3 eqid 2735 . 2 (+g𝑅) = (+g𝑅)
4 eqid 2735 . 2 (+g𝑃) = (+g𝑃)
5 rng2idlring.r . . 3 (𝜑𝑅 ∈ Rng)
6 rnggrp 20176 . . 3 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
75, 6syl 17 . 2 (𝜑𝑅 ∈ Grp)
8 rng2idlring.i . . . 4 (𝜑𝐼 ∈ (2Ideal‘𝑅))
9 rng2idlring.j . . . 4 𝐽 = (𝑅s 𝐼)
10 rng2idlring.u . . . 4 (𝜑𝐽 ∈ Ring)
11 rng2idlring.t . . . 4 · = (.r𝑅)
12 rng2idlring.1 . . . 4 1 = (1r𝐽)
13 rngqiprngim.g . . . 4 = (𝑅 ~QG 𝐼)
14 rngqiprngim.q . . . 4 𝑄 = (𝑅 /s )
15 rngqiprngim.c . . . 4 𝐶 = (Base‘𝑄)
16 rngqiprngim.p . . . 4 𝑃 = (𝑄 ×s 𝐽)
175, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16rngqiprng 21324 . . 3 (𝜑𝑃 ∈ Rng)
18 rnggrp 20176 . . 3 (𝑃 ∈ Rng → 𝑃 ∈ Grp)
1917, 18syl 17 . 2 (𝜑𝑃 ∈ Grp)
20 rngqiprngim.f . . . 4 𝐹 = (𝑥𝐵 ↦ ⟨[𝑥] , ( 1 · 𝑥)⟩)
215, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20rngqiprngimf 21325 . . 3 (𝜑𝐹:𝐵⟶(𝐶 × 𝐼))
225, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16rngqipbas 21323 . . . 4 (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
2322feq3d 6724 . . 3 (𝜑 → (𝐹:𝐵⟶(Base‘𝑃) ↔ 𝐹:𝐵⟶(𝐶 × 𝐼)))
2421, 23mpbird 257 . 2 (𝜑𝐹:𝐵⟶(Base‘𝑃))
25 ringrng 20299 . . . . . . . . 9 (𝐽 ∈ Ring → 𝐽 ∈ Rng)
2610, 25syl 17 . . . . . . . 8 (𝜑𝐽 ∈ Rng)
279, 26eqeltrrid 2844 . . . . . . 7 (𝜑 → (𝑅s 𝐼) ∈ Rng)
285, 8, 27rng2idlnsg 21294 . . . . . 6 (𝜑𝐼 ∈ (NrmSGrp‘𝑅))
2928, 1, 13, 14ecqusaddd 19223 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → [(𝑎(+g𝑅)𝑏)] = ([𝑎] (+g𝑄)[𝑏] ))
305, 8, 9, 10, 1, 11, 12rngqiprngghmlem3 21317 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( 1 · (𝑎(+g𝑅)𝑏)) = (( 1 · 𝑎)(+g𝐽)( 1 · 𝑏)))
3129, 30opeq12d 4886 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ⟨[(𝑎(+g𝑅)𝑏)] , ( 1 · (𝑎(+g𝑅)𝑏))⟩ = ⟨([𝑎] (+g𝑄)[𝑏] ), (( 1 · 𝑎)(+g𝐽)( 1 · 𝑏))⟩)
32 eqid 2735 . . . . 5 (Base‘𝑄) = (Base‘𝑄)
33 eqid 2735 . . . . 5 (Base‘𝐽) = (Base‘𝐽)
3414ovexi 7465 . . . . . 6 𝑄 ∈ V
3534a1i 11 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑄 ∈ V)
3610adantr 480 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝐽 ∈ Ring)
37 simpl 482 . . . . . 6 ((𝑎𝐵𝑏𝐵) → 𝑎𝐵)
3813, 14, 1, 32quseccl0 19216 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑎𝐵) → [𝑎] ∈ (Base‘𝑄))
395, 37, 38syl2an 596 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → [𝑎] ∈ (Base‘𝑄))
405, 8, 9, 10, 1, 11, 12rngqiprngghmlem1 21315 . . . . . 6 ((𝜑𝑎𝐵) → ( 1 · 𝑎) ∈ (Base‘𝐽))
4140adantrr 717 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( 1 · 𝑎) ∈ (Base‘𝐽))
42 simpr 484 . . . . . 6 ((𝑎𝐵𝑏𝐵) → 𝑏𝐵)
4313, 14, 1, 32quseccl0 19216 . . . . . 6 ((𝑅 ∈ Rng ∧ 𝑏𝐵) → [𝑏] ∈ (Base‘𝑄))
445, 42, 43syl2an 596 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → [𝑏] ∈ (Base‘𝑄))
455, 8, 9, 10, 1, 11, 12rngqiprngghmlem1 21315 . . . . . 6 ((𝜑𝑏𝐵) → ( 1 · 𝑏) ∈ (Base‘𝐽))
4645adantrl 716 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( 1 · 𝑏) ∈ (Base‘𝐽))
4728, 1, 13, 14ecqusaddcl 19224 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ([𝑎] (+g𝑄)[𝑏] ) ∈ (Base‘𝑄))
485, 8, 9, 10, 1, 11, 12rngqiprngghmlem2 21316 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( 1 · 𝑎)(+g𝐽)( 1 · 𝑏)) ∈ (Base‘𝐽))
49 eqid 2735 . . . . 5 (+g𝑄) = (+g𝑄)
50 eqid 2735 . . . . 5 (+g𝐽) = (+g𝐽)
5116, 32, 33, 35, 36, 39, 41, 44, 46, 47, 48, 49, 50, 4xpsadd 17621 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (⟨[𝑎] , ( 1 · 𝑎)⟩(+g𝑃)⟨[𝑏] , ( 1 · 𝑏)⟩) = ⟨([𝑎] (+g𝑄)[𝑏] ), (( 1 · 𝑎)(+g𝐽)( 1 · 𝑏))⟩)
5231, 51eqtr4d 2778 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ⟨[(𝑎(+g𝑅)𝑏)] , ( 1 · (𝑎(+g𝑅)𝑏))⟩ = (⟨[𝑎] , ( 1 · 𝑎)⟩(+g𝑃)⟨[𝑏] , ( 1 · 𝑏)⟩))
535adantr 480 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑅 ∈ Rng)
5437adantl 481 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
5542adantl 481 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
561, 3rngacl 20180 . . . . 5 ((𝑅 ∈ Rng ∧ 𝑎𝐵𝑏𝐵) → (𝑎(+g𝑅)𝑏) ∈ 𝐵)
5753, 54, 55, 56syl3anc 1370 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑅)𝑏) ∈ 𝐵)
585, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20rngqiprngimfv 21326 . . . 4 ((𝜑 ∧ (𝑎(+g𝑅)𝑏) ∈ 𝐵) → (𝐹‘(𝑎(+g𝑅)𝑏)) = ⟨[(𝑎(+g𝑅)𝑏)] , ( 1 · (𝑎(+g𝑅)𝑏))⟩)
5957, 58syldan 591 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹‘(𝑎(+g𝑅)𝑏)) = ⟨[(𝑎(+g𝑅)𝑏)] , ( 1 · (𝑎(+g𝑅)𝑏))⟩)
605, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20rngqiprngimfv 21326 . . . . 5 ((𝜑𝑎𝐵) → (𝐹𝑎) = ⟨[𝑎] , ( 1 · 𝑎)⟩)
6160adantrr 717 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑎) = ⟨[𝑎] , ( 1 · 𝑎)⟩)
625, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20rngqiprngimfv 21326 . . . . 5 ((𝜑𝑏𝐵) → (𝐹𝑏) = ⟨[𝑏] , ( 1 · 𝑏)⟩)
6362adantrl 716 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹𝑏) = ⟨[𝑏] , ( 1 · 𝑏)⟩)
6461, 63oveq12d 7449 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝐹𝑎)(+g𝑃)(𝐹𝑏)) = (⟨[𝑎] , ( 1 · 𝑎)⟩(+g𝑃)⟨[𝑏] , ( 1 · 𝑏)⟩))
6552, 59, 643eqtr4d 2785 . 2 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝐹‘(𝑎(+g𝑅)𝑏)) = ((𝐹𝑎)(+g𝑃)(𝐹𝑏)))
661, 2, 3, 4, 7, 19, 24, 65isghmd 19256 1 (𝜑𝐹 ∈ (𝑅 GrpHom 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cmpt 5231   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  [cec 8742  Basecbs 17245  s cress 17274  +gcplusg 17298  .rcmulr 17299   /s cqus 17552   ×s cxps 17553  Grpcgrp 18964   ~QG cqg 19153   GrpHom cghm 19243  Rngcrng 20170  1rcur 20199  Ringcrg 20251  2Idealc2idl 21277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-prds 17494  df-imas 17555  df-qus 17556  df-xps 17557  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-nsg 19155  df-eqg 19156  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-subrng 20563  df-lss 20948  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-2idl 21278
This theorem is referenced by:  rngqiprngimf1  21328  rngqiprngho  21331
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