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 2739	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		eqid 2739	. 2 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2739	. 2 ⊢ (+g‘𝑃) = (+g‘𝑃)
5		rng2idlring.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
6		rnggrp 20131	. . 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 20131	. . 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 6641	. . 3 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝑃) ↔ 𝐹:𝐵⟶(𝐶 × 𝐼)))
24	21, 23	mpbird 258	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝑃))
25		ringrng 20258	. . . . . . . . 9 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
26	10, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Rng)
27	9, 26	eqeltrrid 2844	. . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
28	5, 8, 27	rng2idlnsg 21260	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
29	28, 1, 13, 14	ecqusaddd 19159	. . . . 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 4813	. . . 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 2739	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
33		eqid 2739	. . . . 5 ⊢ (Base‘𝐽) = (Base‘𝐽)
34	14	ovexi 7391	. . . . . 6 ⊢ 𝑄 ∈ V
35	34	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑄 ∈ V)
36	10	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝐽 ∈ Ring)
37		simpl 483	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐵)
38	13, 14, 1, 32	quseccl0 19152	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵) → [𝑎] ∼ ∈ (Base‘𝑄))
39	5, 37, 38	syl2an 602	. . . . 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 723	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑎) ∈ (Base‘𝐽))
42		simpr 485	. . . . . 6 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
43	13, 14, 1, 32	quseccl0 19152	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑏 ∈ 𝐵) → [𝑏] ∼ ∈ (Base‘𝑄))
44	5, 42, 43	syl2an 602	. . . . 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 722	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑏) ∈ (Base‘𝐽))
47	28, 1, 13, 14	ecqusaddcl 19160	. . . . 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 2739	. . . . 5 ⊢ (+g‘𝑄) = (+g‘𝑄)
50		eqid 2739	. . . . 5 ⊢ (+g‘𝐽) = (+g‘𝐽)
51	16, 32, 33, 35, 36, 39, 41, 44, 46, 47, 48, 49, 50, 4	xpsadd 17530	. . . 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 2777	. . 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 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑅 ∈ Rng)
54	37	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
55	42	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
56	1, 3	rngacl 20135	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) ∈ 𝐵)
57	53, 54, 55, 56	syl3anc 1379	. . . 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 597	. . 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 723	. . . 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 722	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
64	61, 63	oveq12d 7375	. . 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 2784	. 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 19192	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4562   ↦ cmpt 5154   × cxp 5617  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357  [cec 8632  Basecbs 17171   ↾_s cress 17192  +_gcplusg 17212  ._rcmulr 17213   /_s cqus 17461   ×_s cxps 17462  Grpcgrp 18901   ~_QG cqg 19090   GrpHom cghm 19179  Rngcrng 20125  1_rcur 20154  Ringcrg 20206  2Idealc2idl 21243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-tpos 8167  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-er 8634  df-ec 8636  df-qs 8640  df-map 8766  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9346  df-inf 9347  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-z 12517  df-dec 12637  df-uz 12781  df-fz 13454  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-ress 17193  df-plusg 17225  df-mulr 17226  df-sca 17228  df-vsca 17229  df-ip 17230  df-tset 17231  df-ple 17232  df-ds 17234  df-hom 17236  df-cco 17237  df-0g 17396  df-prds 17402  df-imas 17464  df-qus 17465  df-xps 17466  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18904  df-minusg 18905  df-sbg 18906  df-subg 19091  df-nsg 19092  df-eqg 19093  df-ghm 19180  df-cmn 19749  df-abl 19750  df-mgp 20114  df-rng 20126  df-ur 20155  df-ring 20208  df-oppr 20309  df-subrng 20519  df-lss 20923  df-sra 21164  df-rgmod 21165  df-lidl 21202  df-2idl 21244

This theorem is referenced by:  rngqiprngimf1  21294  rngqiprngho  21297