Theorem rngqiprngghm 21209

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 2729	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		eqid 2729	. 2 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2729	. 2 ⊢ (+g‘𝑃) = (+g‘𝑃)
5		rng2idlring.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
6		rnggrp 20067	. . 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 21206	. . 3 ⊢ (𝜑 → 𝑃 ∈ Rng)
18		rnggrp 20067	. . 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 21207	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼))
22	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	rngqipbas 21205	. . . 4 ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
23	22	feq3d 6673	. . 3 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝑃) ↔ 𝐹:𝐵⟶(𝐶 × 𝐼)))
24	21, 23	mpbird 257	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝑃))
25		ringrng 20194	. . . . . . . . 9 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
26	10, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Rng)
27	9, 26	eqeltrrid 2833	. . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
28	5, 8, 27	rng2idlnsg 21176	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
29	28, 1, 13, 14	ecqusaddd 19124	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎(+g‘𝑅)𝑏)] ∼ = ([𝑎] ∼ (+g‘𝑄)[𝑏] ∼ ))
30	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem3 21199	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · (𝑎(+g‘𝑅)𝑏)) = (( 1 · 𝑎)(+g‘𝐽)( 1 · 𝑏)))
31	29, 30	opeq12d 4845	. . . 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 2729	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
33		eqid 2729	. . . . 5 ⊢ (Base‘𝐽) = (Base‘𝐽)
34	14	ovexi 7421	. . . . . 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 19117	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵) → [𝑎] ∼ ∈ (Base‘𝑄))
39	5, 37, 38	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑎] ∼ ∈ (Base‘𝑄))
40	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem1 21197	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ( 1 · 𝑎) ∈ (Base‘𝐽))
41	40	adantrr 717	. . . . 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 19117	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑏 ∈ 𝐵) → [𝑏] ∼ ∈ (Base‘𝑄))
44	5, 42, 43	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑏] ∼ ∈ (Base‘𝑄))
45	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem1 21197	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ( 1 · 𝑏) ∈ (Base‘𝐽))
46	45	adantrl 716	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑏) ∈ (Base‘𝐽))
47	28, 1, 13, 14	ecqusaddcl 19125	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (+g‘𝑄)[𝑏] ∼ ) ∈ (Base‘𝑄))
48	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem2 21198	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(+g‘𝐽)( 1 · 𝑏)) ∈ (Base‘𝐽))
49		eqid 2729	. . . . 5 ⊢ (+g‘𝑄) = (+g‘𝑄)
50		eqid 2729	. . . . 5 ⊢ (+g‘𝐽) = (+g‘𝐽)
51	16, 32, 33, 35, 36, 39, 41, 44, 46, 47, 48, 49, 50, 4	xpsadd 17537	. . . 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 2767	. . 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 20071	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) ∈ 𝐵)
57	53, 54, 55, 56	syl3anc 1373	. . . 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 21208	. . . 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 591	. . 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 21208	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
61	60	adantrr 717	. . . 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 21208	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
63	62	adantrl 716	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
64	61, 63	oveq12d 7405	. . 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 2774	. 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 19157	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595   ↦ cmpt 5188   × cxp 5636  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  [cec 8669  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  ._rcmulr 17221   /_s cqus 17468   ×_s cxps 17469  Grpcgrp 18865   ~_QG cqg 19054   GrpHom cghm 19144  Rngcrng 20061  1_rcur 20090  Ringcrg 20142  2Idealc2idl 21159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-ec 8673  df-qs 8677  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-prds 17410  df-imas 17471  df-qus 17472  df-xps 17473  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-sbg 18870  df-subg 19055  df-nsg 19056  df-eqg 19057  df-ghm 19145  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-oppr 20246  df-subrng 20455  df-lss 20838  df-sra 21080  df-rgmod 21081  df-lidl 21118  df-2idl 21160

This theorem is referenced by:  rngqiprngimf1  21210  rngqiprngho  21213