Theorem rngqiprngghm 21422

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 2770	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		eqid 2770	. 2 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2770	. 2 ⊢ (+g‘𝑃) = (+g‘𝑃)
5		rng2idlring.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
6		rnggrp 20239	. . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
7	5, 6	syl 18	. 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 21419	. . 3 ⊢ (𝜑 → 𝑃 ∈ Rng)
18		rnggrp 20239	. . 3 ⊢ (𝑃 ∈ Rng → 𝑃 ∈ Grp)
19	17, 18	syl 18	. 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 21420	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼))
22	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	rngqipbas 21418	. . . 4 ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
23	22	feq3d 6694	. . 3 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝑃) ↔ 𝐹:𝐵⟶(𝐶 × 𝐼)))
24	21, 23	mpbird 260	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝑃))
25		ringrng 20371	. . . . . . . . 9 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
26	10, 25	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Rng)
27	9, 26	eqeltrrid 2875	. . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
28	5, 8, 27	rng2idlnsg 21388	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
29	28, 1, 13, 14	ecqusaddd 19266	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎(+g‘𝑅)𝑏)] ∼ = ([𝑎] ∼ (+g‘𝑄)[𝑏] ∼ ))
30	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem3 21412	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · (𝑎(+g‘𝑅)𝑏)) = (( 1 · 𝑎)(+g‘𝐽)( 1 · 𝑏)))
31	29, 30	opeq12d 4851	. . . 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 2770	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
33		eqid 2770	. . . . 5 ⊢ (Base‘𝐽) = (Base‘𝐽)
34	14	ovexi 7448	. . . . . 6 ⊢ 𝑄 ∈ V
35	34	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑄 ∈ V)
36	10	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝐽 ∈ Ring)
37		simpl 487	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐵)
38	13, 14, 1, 32	quseccl0 19259	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵) → [𝑎] ∼ ∈ (Base‘𝑄))
39	5, 37, 38	syl2an 607	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑎] ∼ ∈ (Base‘𝑄))
40	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem1 21410	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ( 1 · 𝑎) ∈ (Base‘𝐽))
41	40	adantrr 729	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑎) ∈ (Base‘𝐽))
42		simpr 489	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
43	13, 14, 1, 32	quseccl0 19259	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑏 ∈ 𝐵) → [𝑏] ∼ ∈ (Base‘𝑄))
44	5, 42, 43	syl2an 607	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑏] ∼ ∈ (Base‘𝑄))
45	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem1 21410	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ( 1 · 𝑏) ∈ (Base‘𝐽))
46	45	adantrl 728	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑏) ∈ (Base‘𝐽))
47	28, 1, 13, 14	ecqusaddcl 19267	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (+g‘𝑄)[𝑏] ∼ ) ∈ (Base‘𝑄))
48	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem2 21411	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(+g‘𝐽)( 1 · 𝑏)) ∈ (Base‘𝐽))
49		eqid 2770	. . . . 5 ⊢ (+g‘𝑄) = (+g‘𝑄)
50		eqid 2770	. . . . 5 ⊢ (+g‘𝐽) = (+g‘𝐽)
51	16, 32, 33, 35, 36, 39, 41, 44, 46, 47, 48, 49, 50, 4	xpsadd 17631	. . . 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 2808	. . 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 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑅 ∈ Rng)
54	37	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
55	42	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
56	1, 3	rngacl 20243	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) ∈ 𝐵)
57	53, 54, 55, 56	syl3anc 1396	. . . 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 21421	. . . 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 602	. . 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 21421	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
61	60	adantrr 729	. . . 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 21421	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
63	62	adantrl 728	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
64	61, 63	oveq12d 7432	. . 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 2815	. 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 19298	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2150  Vcvv 3462  ⟨cop 4600   ↦ cmpt 5197   × cxp 5663  ⟶wf 6536  ‘cfv 6540  (class class class)co 7414  [cec 8695  Basecbs 17272   ↾_s cress 17293  +_gcplusg 17313  ._rcmulr 17314   /_s cqus 17562   ×_s cxps 17563  Grpcgrp 19003   ~_QG cqg 19191   GrpHom cghm 19286  Rngcrng 20233  1_rcur 20266  Ringcrg 20318  2Idealc2idl 21371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736  ax-cnex 11159  ax-resscn 11160  ax-1cn 11161  ax-icn 11162  ax-addcl 11163  ax-addrcl 11164  ax-mulcl 11165  ax-mulrcl 11166  ax-mulcom 11167  ax-addass 11168  ax-mulass 11169  ax-distr 11170  ax-i2m1 11171  ax-1ne0 11172  ax-1rid 11173  ax-rnegex 11174  ax-rrecex 11175  ax-cnre 11176  ax-pre-lttri 11177  ax-pre-lttrn 11178  ax-pre-ltadd 11179  ax-pre-mulgt0 11180

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-nel 3072  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7866  df-1st 7989  df-2nd 7990  df-tpos 8225  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-2o 8457  df-er 8697  df-ec 8699  df-qs 8703  df-map 8829  df-ixp 8899  df-en 8947  df-dom 8948  df-sdom 8949  df-fin 8950  df-sup 9405  df-inf 9406  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11446  df-neg 11447  df-nn 12237  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12508  df-z 12595  df-dec 12715  df-uz 12866  df-fz 13539  df-struct 17210  df-sets 17227  df-slot 17245  df-ndx 17257  df-base 17273  df-ress 17294  df-plusg 17326  df-mulr 17327  df-sca 17329  df-vsca 17330  df-ip 17331  df-tset 17332  df-ple 17333  df-ds 17335  df-hom 17337  df-cco 17338  df-0g 17497  df-prds 17503  df-imas 17565  df-qus 17566  df-xps 17567  df-mgm 18701  df-sgrp 18780  df-mnd 18796  df-grp 19006  df-minusg 19007  df-sbg 19008  df-subg 19192  df-nsg 19193  df-eqg 19194  df-ghm 19287  df-cmn 19855  df-abl 19856  df-mgp 20220  df-rng 20234  df-ur 20267  df-ring 20320  df-oppr 20422  df-subrng 20634  df-lss 21036  df-sra 21277  df-rgmod 21278  df-lidl 21315  df-2idl 21372

This theorem is referenced by:  rngqiprngimf1  21423  rngqiprngho  21426