Theorem rngqiprngghm 21293

Description: 𝐹 is a homomorphism of the additive groups of non-unital rings. (Contributed by AV, 24-Feb-2025.)

Hypotheses

Ref	Expression
rng2idlring.r	⊢ (𝜑 → 𝑅 ∈ Rng)
rng2idlring.i	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
rng2idlring.j	⊢ 𝐽 = (𝑅 ↾s 𝐼)
rng2idlring.u	⊢ (𝜑 → 𝐽 ∈ Ring)
rng2idlring.b	⊢ 𝐵 = (Base‘𝑅)
rng2idlring.t	⊢ · = (.r‘𝑅)
rng2idlring.1	⊢ 1 = (1r‘𝐽)
rngqiprngim.g	⊢ ∼ = (𝑅 ~QG 𝐼)
rngqiprngim.q	⊢ 𝑄 = (𝑅 /s ∼ )
rngqiprngim.c	⊢ 𝐶 = (Base‘𝑄)
rngqiprngim.p	⊢ 𝑃 = (𝑄 ×s 𝐽)
rngqiprngim.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)

Assertion

Ref	Expression
rngqiprngghm	⊢ (𝜑 → 𝐹 ∈ (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.

Step	Hyp	Ref	Expression
1		rng2idlring.b	. 2 ⊢ 𝐵 = (Base‘𝑅)
2		eqid 2737	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		eqid 2737	. 2 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2737	. 2 ⊢ (+g‘𝑃) = (+g‘𝑃)
5		rng2idlring.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
6		rnggrp 20134	. . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → 𝑅 ∈ Grp)
8		rng2idlring.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
9		rng2idlring.j	. . . 4 ⊢ 𝐽 = (𝑅 ↾s 𝐼)
10		rng2idlring.u	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Ring)
11		rng2idlring.t	. . . 4 ⊢ · = (.r‘𝑅)
12		rng2idlring.1	. . . 4 ⊢ 1 = (1r‘𝐽)
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)
17	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	rngqiprng 21290	. . 3 ⊢ (𝜑 → 𝑃 ∈ Rng)
18		rnggrp 20134	. . 3 ⊢ (𝑃 ∈ Rng → 𝑃 ∈ Grp)
19	17, 18	syl 17	. 2 ⊢ (𝜑 → 𝑃 ∈ Grp)
20		rngqiprngim.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)
21	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20	rngqiprngimf 21291	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼))
22	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	rngqipbas 21289	. . . 4 ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
23	22	feq3d 6649	. . 3 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝑃) ↔ 𝐹:𝐵⟶(𝐶 × 𝐼)))
24	21, 23	mpbird 257	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝑃))
25		ringrng 20261	. . . . . . . . 9 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
26	10, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Rng)
27	9, 26	eqeltrrid 2842	. . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
28	5, 8, 27	rng2idlnsg 21260	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
29	28, 1, 13, 14	ecqusaddd 19162	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎(+g‘𝑅)𝑏)] ∼ = ([𝑎] ∼ (+g‘𝑄)[𝑏] ∼ ))
30	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem3 21283	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · (𝑎(+g‘𝑅)𝑏)) = (( 1 · 𝑎)(+g‘𝐽)( 1 · 𝑏)))
31	29, 30	opeq12d 4825	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ⟨[(𝑎(+g‘𝑅)𝑏)] ∼ , ( 1 · (𝑎(+g‘𝑅)𝑏))⟩ = ⟨([𝑎] ∼ (+g‘𝑄)[𝑏] ∼ ), (( 1 · 𝑎)(+g‘𝐽)( 1 · 𝑏))⟩)
32		eqid 2737	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
33		eqid 2737	. . . . 5 ⊢ (Base‘𝐽) = (Base‘𝐽)
34	14	ovexi 7396	. . . . . 6 ⊢ 𝑄 ∈ V
35	34	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑄 ∈ V)
36	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝐽 ∈ Ring)
37		simpl 482	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐵)
38	13, 14, 1, 32	quseccl0 19155	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵) → [𝑎] ∼ ∈ (Base‘𝑄))
39	5, 37, 38	syl2an 597	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑎] ∼ ∈ (Base‘𝑄))
40	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem1 21281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ( 1 · 𝑎) ∈ (Base‘𝐽))
41	40	adantrr 718	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑎) ∈ (Base‘𝐽))
42		simpr 484	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
43	13, 14, 1, 32	quseccl0 19155	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑏 ∈ 𝐵) → [𝑏] ∼ ∈ (Base‘𝑄))
44	5, 42, 43	syl2an 597	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑏] ∼ ∈ (Base‘𝑄))
45	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem1 21281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ( 1 · 𝑏) ∈ (Base‘𝐽))
46	45	adantrl 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑏) ∈ (Base‘𝐽))
47	28, 1, 13, 14	ecqusaddcl 19163	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (+g‘𝑄)[𝑏] ∼ ) ∈ (Base‘𝑄))
48	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem2 21282	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(+g‘𝐽)( 1 · 𝑏)) ∈ (Base‘𝐽))
49		eqid 2737	. . . . 5 ⊢ (+g‘𝑄) = (+g‘𝑄)
50		eqid 2737	. . . . 5 ⊢ (+g‘𝐽) = (+g‘𝐽)
51	16, 32, 33, 35, 36, 39, 41, 44, 46, 47, 48, 49, 50, 4	xpsadd 17533	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩(+g‘𝑃)⟨[𝑏] ∼ , ( 1 · 𝑏)⟩) = ⟨([𝑎] ∼ (+g‘𝑄)[𝑏] ∼ ), (( 1 · 𝑎)(+g‘𝐽)( 1 · 𝑏))⟩)
52	31, 51	eqtr4d 2775	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ⟨[(𝑎(+g‘𝑅)𝑏)] ∼ , ( 1 · (𝑎(+g‘𝑅)𝑏))⟩ = (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩(+g‘𝑃)⟨[𝑏] ∼ , ( 1 · 𝑏)⟩))
53	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑅 ∈ Rng)
54	37	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
55	42	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
56	1, 3	rngacl 20138	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) ∈ 𝐵)
57	53, 54, 55, 56	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑅)𝑏) ∈ 𝐵)
58	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20	rngqiprngimfv 21292	. . . 4 ⊢ ((𝜑 ∧ (𝑎(+g‘𝑅)𝑏) ∈ 𝐵) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ⟨[(𝑎(+g‘𝑅)𝑏)] ∼ , ( 1 · (𝑎(+g‘𝑅)𝑏))⟩)
59	57, 58	syldan 592	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ⟨[(𝑎(+g‘𝑅)𝑏)] ∼ , ( 1 · (𝑎(+g‘𝑅)𝑏))⟩)
60	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20	rngqiprngimfv 21292	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
61	60	adantrr 718	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
62	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 20	rngqiprngimfv 21292	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
63	62	adantrl 717	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
64	61, 63	oveq12d 7380	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝐹‘𝑎)(+g‘𝑃)(𝐹‘𝑏)) = (⟨[𝑎] ∼ , ( 1 · 𝑎)⟩(+g‘𝑃)⟨[𝑏] ∼ , ( 1 · 𝑏)⟩))
65	52, 59, 64	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑃)(𝐹‘𝑏)))
66	1, 2, 3, 4, 7, 19, 24, 65	isghmd 19195	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   ↦ cmpt 5167   × cxp 5624  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362  [cec 8636  Basecbs 17174   ↾_s cress 17195  +_gcplusg 17215  ._rcmulr 17216   /_s cqus 17464   ×_s cxps 17465  Grpcgrp 18904   ~_QG cqg 19093   GrpHom cghm 19182  Rngcrng 20128  1_rcur 20157  Ringcrg 20209  2Idealc2idl 21243

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-tpos 8171  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-er 8638  df-ec 8640  df-qs 8644  df-map 8770  df-ixp 8841  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-sca 17231  df-vsca 17232  df-ip 17233  df-tset 17234  df-ple 17235  df-ds 17237  df-hom 17239  df-cco 17240  df-0g 17399  df-prds 17405  df-imas 17467  df-qus 17468  df-xps 17469  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18907  df-minusg 18908  df-sbg 18909  df-subg 19094  df-nsg 19095  df-eqg 19096  df-ghm 19183  df-cmn 19752  df-abl 19753  df-mgp 20117  df-rng 20129  df-ur 20158  df-ring 20211  df-oppr 20312  df-subrng 20518  df-lss 20922  df-sra 21164  df-rgmod 21165  df-lidl 21202  df-2idl 21244

This theorem is referenced by:  rngqiprngimf1  21294  rngqiprngho  21297