Theorem rhmqusspan 42679

Description: Ring homomorphism out of a quotient given an ideal spanned by a singleton. (Contributed by metakunt, 7-Jun-2025.)

Hypotheses

Ref	Expression
rhmqusspan.1	⊢ 0 = (0g‘𝐻)
rhmqusspan.2	⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))
rhmqusspan.3	⊢ 𝐾 = (◡𝐹 “ { 0 })
rhmqusspan.4	⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
rhmqusspan.5	⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
rhmqusspan.6	⊢ (𝜑 → 𝐺 ∈ CRing)
rhmqusspan.7	⊢ 𝑁 = ((RSpan‘𝐺)‘{𝑋})
rhmqusspan.8	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))
rhmqusspan.9	⊢ (𝜑 → (𝐹‘𝑋) = 0 )

Assertion

Ref	Expression
rhmqusspan	⊢ (𝜑 → (𝐽 ∈ (𝑄 RingHom 𝐻) ∧ ∀𝑔 ∈ (Base‘𝐺)(𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔)))

Distinct variable groups:   𝐹,𝑞   𝐺,𝑞   𝐻,𝑞   𝐽,𝑞   𝐾,𝑞   𝑁,𝑞   𝑄,𝑞   𝜑,𝑔,𝑞

Allowed substitution hints:   𝑄(𝑔)   𝐹(𝑔)   𝐺(𝑔)   𝐻(𝑔)   𝐽(𝑔)   𝐾(𝑔)   𝑁(𝑔)   𝑋(𝑔,𝑞)   0 (𝑔,𝑞)

Proof of Theorem rhmqusspan

Dummy variables 𝑥 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmqusspan.1	. . 3 ⊢ 0 = (0g‘𝐻)
2		rhmqusspan.2	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))
3		rhmqusspan.3	. . 3 ⊢ 𝐾 = (◡𝐹 “ { 0 })
4		rhmqusspan.4	. . 3 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
5		rhmqusspan.5	. . 3 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
6		rhmqusspan.6	. . 3 ⊢ (𝜑 → 𝐺 ∈ CRing)
7		rhmqusspan.7	. . . 4 ⊢ 𝑁 = ((RSpan‘𝐺)‘{𝑋})
8	6	crngringd 20219	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ Ring)
9		rhmqusspan.8	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))
10		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐺) = (Base‘𝐺)
11		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (RSpan‘𝐺) = (RSpan‘𝐺)
12		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (∥r‘𝐺) = (∥r‘𝐺)
13	10, 11, 12	rspsn 21327	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Ring ∧ 𝑋 ∈ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) = {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦})
14	8, 9, 13	syl2anc 590	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) = {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦})
15	14	eleq2d 2825	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ ((RSpan‘𝐺)‘{𝑋}) ↔ 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}))
16	15	biimpd 230	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ((RSpan‘𝐺)‘{𝑋}) → 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}))
17	16	imp 407	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦})
18		vex 3435	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
19	18	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑥 ∈ V)
20		breq2 5077	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑋(∥r‘𝐺)𝑦 ↔ 𝑋(∥r‘𝐺)𝑥))
21	20	elabg 3614	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → (𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦} ↔ 𝑋(∥r‘𝐺)𝑥))
22	21	biimpd 230	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ V → (𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦} → 𝑋(∥r‘𝐺)𝑥))
23	19, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦} → 𝑋(∥r‘𝐺)𝑥))
24	23	imp 407	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}) → 𝑋(∥r‘𝐺)𝑥)
25		eqid 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (.r‘𝐺) = (.r‘𝐺)
26	10, 12, 25	dvdsr 20334	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋(∥r‘𝐺)𝑥 ↔ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥))
27	26	bilani 505	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑋(∥r‘𝐺)𝑥) → (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥))
28		fveq2 6828	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧(.r‘𝐺)𝑋) = 𝑥 → (𝐹‘(𝑧(.r‘𝐺)𝑋)) = (𝐹‘𝑥))
29	28	eqcomd 2745	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧(.r‘𝐺)𝑋) = 𝑥 → (𝐹‘𝑥) = (𝐹‘(𝑧(.r‘𝐺)𝑋)))
30	29	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) ∧ (𝑧(.r‘𝐺)𝑋) = 𝑥) → (𝐹‘𝑥) = (𝐹‘(𝑧(.r‘𝐺)𝑋)))
31	2	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
32		simpr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝑧 ∈ (Base‘𝐺))
33	9	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝑋 ∈ (Base‘𝐺))
34		eqid 2739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (.r‘𝐻) = (.r‘𝐻)
35	10, 25, 34	rhmmul 20458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 ∈ (𝐺 RingHom 𝐻) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑋 ∈ (Base‘𝐺)) → (𝐹‘(𝑧(.r‘𝐺)𝑋)) = ((𝐹‘𝑧)(.r‘𝐻)(𝐹‘𝑋)))
36	31, 32, 33, 35	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝐹‘(𝑧(.r‘𝐺)𝑋)) = ((𝐹‘𝑧)(.r‘𝐻)(𝐹‘𝑋)))
37		rhmqusspan.9	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐹‘𝑋) = 0 )
38	37	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝐹‘𝑋) = 0 )
39	38	oveq2d 7373	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝐹‘𝑧)(.r‘𝐻)(𝐹‘𝑋)) = ((𝐹‘𝑧)(.r‘𝐻) 0 ))
40		rhmrcl2 20449	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
41		ringsrg 20270	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐻 ∈ Ring → 𝐻 ∈ SRing)
42	31, 40, 41	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝐻 ∈ SRing)
43		eqid 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Base‘𝐻) = (Base‘𝐻)
44	10, 43	rhmf 20456	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
45	2, 44	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
46	45	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
47	46	ffvelcdmda 7026	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝐹‘𝑧) ∈ (Base‘𝐻))
48	43, 34, 1	srgrz 20180	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐻 ∈ SRing ∧ (𝐹‘𝑧) ∈ (Base‘𝐻)) → ((𝐹‘𝑧)(.r‘𝐻) 0 ) = 0 )
49	42, 47, 48	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝐹‘𝑧)(.r‘𝐻) 0 ) = 0 )
50	39, 49	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝐹‘𝑧)(.r‘𝐻)(𝐹‘𝑋)) = 0 )
51	36, 50	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝐹‘(𝑧(.r‘𝐺)𝑋)) = 0 )
52	51	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) ∧ (𝑧(.r‘𝐺)𝑋) = 𝑥) → (𝐹‘(𝑧(.r‘𝐺)𝑋)) = 0 )
53	30, 52	eqtrd 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) ∧ (𝑧(.r‘𝐺)𝑋) = 𝑥) → (𝐹‘𝑥) = 0 )
54		nfv 1921	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑧(𝑦(.r‘𝐺)𝑋) = 𝑥
55		nfv 1921	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑦(𝑧(.r‘𝐺)𝑋) = 𝑥
56		oveq1 7364	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → (𝑦(.r‘𝐺)𝑋) = (𝑧(.r‘𝐺)𝑋))
57	56	eqeq1d 2741	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑧 → ((𝑦(.r‘𝐺)𝑋) = 𝑥 ↔ (𝑧(.r‘𝐺)𝑋) = 𝑥))
58	54, 55, 57	cbvrexw 3282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥 ↔ ∃𝑧 ∈ (Base‘𝐺)(𝑧(.r‘𝐺)𝑋) = 𝑥)
59	58	bilani 505	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥) → ∃𝑧 ∈ (Base‘𝐺)(𝑧(.r‘𝐺)𝑋) = 𝑥)
60	59	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) → ∃𝑧 ∈ (Base‘𝐺)(𝑧(.r‘𝐺)𝑋) = 𝑥)
61	53, 60	r19.29a 3147	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) → (𝐹‘𝑥) = 0 )
62	61	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥) → (𝐹‘𝑥) = 0 ))
63	62	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑋(∥r‘𝐺)𝑥) → ((𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥) → (𝐹‘𝑥) = 0 ))
64	27, 63	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋(∥r‘𝐺)𝑥) → (𝐹‘𝑥) = 0 )
65	64	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋(∥r‘𝐺)𝑥 → (𝐹‘𝑥) = 0 ))
66	65	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}) → (𝑋(∥r‘𝐺)𝑥 → (𝐹‘𝑥) = 0 ))
67	24, 66	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}) → (𝐹‘𝑥) = 0 )
68	67	ex 413	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦} → (𝐹‘𝑥) = 0 ))
69	68	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦} → (𝐹‘𝑥) = 0 ))
70	17, 69	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝐹‘𝑥) = 0 )
71		fvexd 6843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝐹‘𝑥) ∈ V)
72		elsng 4570	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ V → ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 ))
73	71, 72	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 ))
74	70, 73	mpbird 258	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝐹‘𝑥) ∈ { 0 })
75	45	ffund 6660	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
76	75	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → Fun 𝐹)
77		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (LIdeal‘𝐺) = (LIdeal‘𝐺)
78	77, 10	lidl1 21227	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Ring → (Base‘𝐺) ∈ (LIdeal‘𝐺))
79	8, 78	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐺) ∈ (LIdeal‘𝐺))
80	9	snssd 4719	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑋} ⊆ (Base‘𝐺))
81	11, 77	rspssp 21233	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Ring ∧ (Base‘𝐺) ∈ (LIdeal‘𝐺) ∧ {𝑋} ⊆ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) ⊆ (Base‘𝐺))
82	8, 79, 80, 81	syl3anc 1379	. . . . . . . . . . 11 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ (Base‘𝐺))
83	82	sselda 3915	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ (Base‘𝐺))
84		fdm 6665	. . . . . . . . . . . 12 ⊢ (𝐹:(Base‘𝐺)⟶(Base‘𝐻) → dom 𝐹 = (Base‘𝐺))
85	45, 84	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = (Base‘𝐺))
86	85	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → dom 𝐹 = (Base‘𝐺))
87	83, 86	eleqtrrd 2842	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ dom 𝐹)
88		fvimacnv 6995	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ { 0 } ↔ 𝑥 ∈ (◡𝐹 “ { 0 })))
89	76, 87, 88	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → ((𝐹‘𝑥) ∈ { 0 } ↔ 𝑥 ∈ (◡𝐹 “ { 0 })))
90	74, 89	mpbid 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ (◡𝐹 “ { 0 }))
91	90	ex 413	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((RSpan‘𝐺)‘{𝑋}) → 𝑥 ∈ (◡𝐹 “ { 0 })))
92	91	ssrdv 3921	. . . . 5 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ (◡𝐹 “ { 0 }))
93	3	eqcomi 2748	. . . . 5 ⊢ (◡𝐹 “ { 0 }) = 𝐾
94	92, 93	sseqtrdi 3955	. . . 4 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ 𝐾)
95	7, 94	eqsstrid 3953	. . 3 ⊢ (𝜑 → 𝑁 ⊆ 𝐾)
96	11, 10, 77	rspcl 21229	. . . . 5 ⊢ ((𝐺 ∈ Ring ∧ {𝑋} ⊆ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) ∈ (LIdeal‘𝐺))
97	8, 80, 96	syl2anc 590	. . . 4 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ∈ (LIdeal‘𝐺))
98	7, 97	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝑁 ∈ (LIdeal‘𝐺))
99	1, 2, 3, 4, 5, 6, 95, 98	rhmqusnsg 21279	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))
100	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
101		rhmghm 20455	. . . . 5 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
102	100, 101	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
103	95	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝑁 ⊆ 𝐾)
104		lidlnsg 21242	. . . . . 6 ⊢ ((𝐺 ∈ Ring ∧ 𝑁 ∈ (LIdeal‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
105	8, 98, 104	syl2anc 590	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
106	105	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
107		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝑔 ∈ (Base‘𝐺))
108	1, 102, 3, 4, 5, 103, 106, 107	ghmqusnsglem1 19247	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → (𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔))
109	108	ralrimiva 3131	. 2 ⊢ (𝜑 → ∀𝑔 ∈ (Base‘𝐺)(𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔))
110	99, 109	jca 516	1 ⊢ (𝜑 → (𝐽 ∈ (𝑄 RingHom 𝐻) ∧ ∀𝑔 ∈ (Base‘𝐺)(𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  {csn 4556  ∪ cuni 4839   class class class wbr 5073   ↦ cmpt 5154  ^◡ccnv 5618  dom cdm 5619   “ cima 5622  Fun wfun 6480  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357  [cec 8632  Basecbs 17171  ._rcmulr 17213  0_gc0g 17394   /_s cqus 17461  NrmSGrpcnsg 19089   ~_QG cqg 19090   GrpHom cghm 19179  SRingcsrg 20159  Ringcrg 20206  CRingccrg 20207  ∥_rcdsr 20326   RingHom crh 20441  LIdealclidl 21200  RSpancrsp 21201

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-int 4879  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-er 8634  df-ec 8636  df-qs 8640  df-map 8766  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-0g 17396  df-imas 17464  df-qus 17465  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-mhm 18743  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-srg 20160  df-ring 20208  df-cring 20209  df-oppr 20309  df-dvdsr 20329  df-rhm 20444  df-subrg 20543  df-lmod 20853  df-lss 20923  df-lsp 20963  df-sra 21164  df-rgmod 21165  df-lidl 21202  df-rsp 21203  df-2idl 21244

This theorem is referenced by:  aks5lem2  42681