Theorem rngqiprngghm 21332

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 2740	. 2 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		eqid 2740	. 2 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		eqid 2740	. 2 ⊢ (+g‘𝑃) = (+g‘𝑃)
5		rng2idlring.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
6		rnggrp 20185	. . 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 21329	. . 3 ⊢ (𝜑 → 𝑃 ∈ Rng)
18		rnggrp 20185	. . 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 21330	. . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼))
22	5, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16	rngqipbas 21328	. . . 4 ⊢ (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
23	22	feq3d 6734	. . 3 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝑃) ↔ 𝐹:𝐵⟶(𝐶 × 𝐼)))
24	21, 23	mpbird 257	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝑃))
25		ringrng 20308	. . . . . . . . 9 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng)
26	10, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Rng)
27	9, 26	eqeltrrid 2849	. . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)
28	5, 8, 27	rng2idlnsg 21299	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
29	28, 1, 13, 14	ecqusaddd 19232	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [(𝑎(+g‘𝑅)𝑏)] ∼ = ([𝑎] ∼ (+g‘𝑄)[𝑏] ∼ ))
30	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem3 21322	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · (𝑎(+g‘𝑅)𝑏)) = (( 1 · 𝑎)(+g‘𝐽)( 1 · 𝑏)))
31	29, 30	opeq12d 4905	. . . 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 2740	. . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄)
33		eqid 2740	. . . . 5 ⊢ (Base‘𝐽) = (Base‘𝐽)
34	14	ovexi 7482	. . . . . 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 19225	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵) → [𝑎] ∼ ∈ (Base‘𝑄))
39	5, 37, 38	syl2an 595	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑎] ∼ ∈ (Base‘𝑄))
40	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem1 21320	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ( 1 · 𝑎) ∈ (Base‘𝐽))
41	40	adantrr 716	. . . . 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 19225	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑏 ∈ 𝐵) → [𝑏] ∼ ∈ (Base‘𝑄))
44	5, 42, 43	syl2an 595	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → [𝑏] ∼ ∈ (Base‘𝑄))
45	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem1 21320	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ( 1 · 𝑏) ∈ (Base‘𝐽))
46	45	adantrl 715	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( 1 · 𝑏) ∈ (Base‘𝐽))
47	28, 1, 13, 14	ecqusaddcl 19233	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ([𝑎] ∼ (+g‘𝑄)[𝑏] ∼ ) ∈ (Base‘𝑄))
48	5, 8, 9, 10, 1, 11, 12	rngqiprngghmlem2 21321	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( 1 · 𝑎)(+g‘𝐽)( 1 · 𝑏)) ∈ (Base‘𝐽))
49		eqid 2740	. . . . 5 ⊢ (+g‘𝑄) = (+g‘𝑄)
50		eqid 2740	. . . . 5 ⊢ (+g‘𝐽) = (+g‘𝐽)
51	16, 32, 33, 35, 36, 39, 41, 44, 46, 47, 48, 49, 50, 4	xpsadd 17634	. . . 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 2783	. . 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 20189	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑅)𝑏) ∈ 𝐵)
57	53, 54, 55, 56	syl3anc 1371	. . . 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 21331	. . . 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 590	. . 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 21331	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐹‘𝑎) = ⟨[𝑎] ∼ , ( 1 · 𝑎)⟩)
61	60	adantrr 716	. . . 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 21331	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
63	62	adantrl 715	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = ⟨[𝑏] ∼ , ( 1 · 𝑏)⟩)
64	61, 63	oveq12d 7466	. . 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 2790	. 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 19265	1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ⟨cop 4654   ↦ cmpt 5249   × cxp 5698  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  [cec 8761  Basecbs 17258   ↾_s cress 17287  +_gcplusg 17311  ._rcmulr 17312   /_s cqus 17565   ×_s cxps 17566  Grpcgrp 18973   ~_QG cqg 19162   GrpHom cghm 19252  Rngcrng 20179  1_rcur 20208  Ringcrg 20260  2Idealc2idl 21282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-ec 8765  df-qs 8769  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17501  df-prds 17507  df-imas 17568  df-qus 17569  df-xps 17570  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163  df-nsg 19164  df-eqg 19165  df-ghm 19253  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-oppr 20360  df-subrng 20572  df-lss 20953  df-sra 21195  df-rgmod 21196  df-lidl 21241  df-2idl 21283

This theorem is referenced by:  rngqiprngimf1  21333  rngqiprngho  21336