Theorem rhmqusspan 41788

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 20198	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ Ring)
9		rhmqusspan.8	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))
10		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐺) = (Base‘𝐺)
11		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (RSpan‘𝐺) = (RSpan‘𝐺)
12		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (∥r‘𝐺) = (∥r‘𝐺)
13	10, 11, 12	rspsn 21240	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Ring ∧ 𝑋 ∈ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) = {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦})
14	8, 9, 13	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) = {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦})
15	14	eleq2d 2811	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ ((RSpan‘𝐺)‘{𝑋}) ↔ 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}))
16	15	biimpd 228	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ((RSpan‘𝐺)‘{𝑋}) → 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}))
17	16	imp 405	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦})
18		vex 3465	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
19	18	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑥 ∈ V)
20		breq2 5153	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑋(∥r‘𝐺)𝑦 ↔ 𝑋(∥r‘𝐺)𝑥))
21	20	elabg 3662	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → (𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦} ↔ 𝑋(∥r‘𝐺)𝑥))
22	21	biimpd 228	. . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}) → 𝑋(∥r‘𝐺)𝑥)
25		eqid 2725	. . . . . . . . . . . . . . . . . . 19 ⊢ (.r‘𝐺) = (.r‘𝐺)
26	10, 12, 25	dvdsr 20313	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋(∥r‘𝐺)𝑥 ↔ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥))
27	26	biimpi 215	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋(∥r‘𝐺)𝑥 → (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥))
28	27	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑋(∥r‘𝐺)𝑥) → (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥))
29		fveq2 6896	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧(.r‘𝐺)𝑋) = 𝑥 → (𝐹‘(𝑧(.r‘𝐺)𝑋)) = (𝐹‘𝑥))
30	29	eqcomd 2731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧(.r‘𝐺)𝑋) = 𝑥 → (𝐹‘𝑥) = (𝐹‘(𝑧(.r‘𝐺)𝑋)))
31	30	adantl 480	. . . . . . . . . . . . . . . . . . . 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)))
32	2	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
33		simpr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝑧 ∈ (Base‘𝐺))
34	9	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝑋 ∈ (Base‘𝐺))
35		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (.r‘𝐻) = (.r‘𝐻)
36	10, 25, 35	rhmmul 20437	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 ∈ (𝐺 RingHom 𝐻) ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑋 ∈ (Base‘𝐺)) → (𝐹‘(𝑧(.r‘𝐺)𝑋)) = ((𝐹‘𝑧)(.r‘𝐻)(𝐹‘𝑋)))
37	32, 33, 34, 36	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . 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)))
38		rhmqusspan.9	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐹‘𝑋) = 0 )
39	38	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝐹‘𝑋) = 0 )
40	39	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . 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 ))
41		rhmrcl2 20428	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
42		ringsrg 20245	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐻 ∈ Ring → 𝐻 ∈ SRing)
43	32, 41, 42	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝐻 ∈ SRing)
44		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Base‘𝐻) = (Base‘𝐻)
45	10, 44	rhmf 20436	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
46	2, 45	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
47	46	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
48	47	ffvelcdmda 7093	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝐹‘𝑧) ∈ (Base‘𝐻))
49	44, 35, 1	srgrz 20159	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐻 ∈ SRing ∧ (𝐹‘𝑧) ∈ (Base‘𝐻)) → ((𝐹‘𝑧)(.r‘𝐻) 0 ) = 0 )
50	43, 48, 49	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝐹‘𝑧)(.r‘𝐻) 0 ) = 0 )
51	40, 50	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝐹‘𝑧)(.r‘𝐻)(𝐹‘𝑋)) = 0 )
52	37, 51	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝐹‘(𝑧(.r‘𝐺)𝑋)) = 0 )
53	52	adantr 479	. . . . . . . . . . . . . . . . . . . 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 )
54	31, 53	eqtrd 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) ∧ (𝑧(.r‘𝐺)𝑋) = 𝑥) → (𝐹‘𝑥) = 0 )
55		nfv 1909	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑧(𝑦(.r‘𝐺)𝑋) = 𝑥
56		nfv 1909	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑦(𝑧(.r‘𝐺)𝑋) = 𝑥
57		oveq1 7426	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 → (𝑦(.r‘𝐺)𝑋) = (𝑧(.r‘𝐺)𝑋))
58	57	eqeq1d 2727	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → ((𝑦(.r‘𝐺)𝑋) = 𝑥 ↔ (𝑧(.r‘𝐺)𝑋) = 𝑥))
59	55, 56, 58	cbvrexw 3294	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥 ↔ ∃𝑧 ∈ (Base‘𝐺)(𝑧(.r‘𝐺)𝑋) = 𝑥)
60	59	biimpi 215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥 → ∃𝑧 ∈ (Base‘𝐺)(𝑧(.r‘𝐺)𝑋) = 𝑥)
61	60	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥) → ∃𝑧 ∈ (Base‘𝐺)(𝑧(.r‘𝐺)𝑋) = 𝑥)
62	61	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) → ∃𝑧 ∈ (Base‘𝐺)(𝑧(.r‘𝐺)𝑋) = 𝑥)
63	54, 62	r19.29a 3151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) → (𝐹‘𝑥) = 0 )
64	63	ex 411	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥) → (𝐹‘𝑥) = 0 ))
65	64	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑋(∥r‘𝐺)𝑥) → ((𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥) → (𝐹‘𝑥) = 0 ))
66	28, 65	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑋(∥r‘𝐺)𝑥) → (𝐹‘𝑥) = 0 )
67	66	ex 411	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋(∥r‘𝐺)𝑥 → (𝐹‘𝑥) = 0 ))
68	67	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}) → (𝑋(∥r‘𝐺)𝑥 → (𝐹‘𝑥) = 0 ))
69	24, 68	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}) → (𝐹‘𝑥) = 0 )
70	69	ex 411	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦} → (𝐹‘𝑥) = 0 ))
71	70	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦} → (𝐹‘𝑥) = 0 ))
72	17, 71	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝐹‘𝑥) = 0 )
73		fvexd 6911	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝐹‘𝑥) ∈ V)
74		elsng 4644	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ V → ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 ))
75	73, 74	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 ))
76	72, 75	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝐹‘𝑥) ∈ { 0 })
77	46	ffund 6727	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
78	77	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → Fun 𝐹)
79		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (LIdeal‘𝐺) = (LIdeal‘𝐺)
80	79, 10	lidl1 21141	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Ring → (Base‘𝐺) ∈ (LIdeal‘𝐺))
81	8, 80	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐺) ∈ (LIdeal‘𝐺))
82	9	snssd 4814	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑋} ⊆ (Base‘𝐺))
83	11, 79	rspssp 21147	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Ring ∧ (Base‘𝐺) ∈ (LIdeal‘𝐺) ∧ {𝑋} ⊆ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) ⊆ (Base‘𝐺))
84	8, 81, 82, 83	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ (Base‘𝐺))
85	84	sselda 3976	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ (Base‘𝐺))
86		fdm 6732	. . . . . . . . . . . 12 ⊢ (𝐹:(Base‘𝐺)⟶(Base‘𝐻) → dom 𝐹 = (Base‘𝐺))
87	46, 86	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = (Base‘𝐺))
88	87	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → dom 𝐹 = (Base‘𝐺))
89	85, 88	eleqtrrd 2828	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ dom 𝐹)
90		fvimacnv 7061	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ { 0 } ↔ 𝑥 ∈ (◡𝐹 “ { 0 })))
91	78, 89, 90	syl2anc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → ((𝐹‘𝑥) ∈ { 0 } ↔ 𝑥 ∈ (◡𝐹 “ { 0 })))
92	76, 91	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ (◡𝐹 “ { 0 }))
93	92	ex 411	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((RSpan‘𝐺)‘{𝑋}) → 𝑥 ∈ (◡𝐹 “ { 0 })))
94	93	ssrdv 3982	. . . . 5 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ (◡𝐹 “ { 0 }))
95	3	eqcomi 2734	. . . . 5 ⊢ (◡𝐹 “ { 0 }) = 𝐾
96	94, 95	sseqtrdi 4027	. . . 4 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ 𝐾)
97	7, 96	eqsstrid 4025	. . 3 ⊢ (𝜑 → 𝑁 ⊆ 𝐾)
98	11, 10, 79	rspcl 21143	. . . . 5 ⊢ ((𝐺 ∈ Ring ∧ {𝑋} ⊆ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) ∈ (LIdeal‘𝐺))
99	8, 82, 98	syl2anc 582	. . . 4 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ∈ (LIdeal‘𝐺))
100	7, 99	eqeltrid 2829	. . 3 ⊢ (𝜑 → 𝑁 ∈ (LIdeal‘𝐺))
101	1, 2, 3, 4, 5, 6, 97, 100	rhmqusnsg 21192	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))
102	2	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
103		rhmghm 20435	. . . . 5 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
104	102, 103	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
105	97	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝑁 ⊆ 𝐾)
106		lidlnsg 21155	. . . . . 6 ⊢ ((𝐺 ∈ Ring ∧ 𝑁 ∈ (LIdeal‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
107	8, 100, 106	syl2anc 582	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
108	107	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
109		simpr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝑔 ∈ (Base‘𝐺))
110	1, 104, 3, 4, 5, 105, 108, 109	ghmqusnsglem1 19243	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → (𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔))
111	110	ralrimiva 3135	. 2 ⊢ (𝜑 → ∀𝑔 ∈ (Base‘𝐺)(𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔))
112	101, 111	jca 510	1 ⊢ (𝜑 → (𝐽 ∈ (𝑄 RingHom 𝐻) ∧ ∀𝑔 ∈ (Base‘𝐺)(𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2702  ∀wral 3050  ∃wrex 3059  Vcvv 3461   ⊆ wss 3944  {csn 4630  ∪ cuni 4909   class class class wbr 5149   ↦ cmpt 5232  ^◡ccnv 5677  dom cdm 5678   “ cima 5681  Fun wfun 6543  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  [cec 8723  Basecbs 17183  ._rcmulr 17237  0_gc0g 17424   /_s cqus 17490  NrmSGrpcnsg 19084   ~_QG cqg 19085   GrpHom cghm 19175  SRingcsrg 20138  Ringcrg 20185  CRingccrg 20186  ∥_rcdsr 20305   RingHom crh 20420  LIdealclidl 21114  RSpancrsp 21115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-ec 8727  df-qs 8731  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-fz 13520  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-0g 17426  df-imas 17493  df-qus 17494  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18743  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19086  df-nsg 19087  df-eqg 19088  df-ghm 19176  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-srg 20139  df-ring 20187  df-cring 20188  df-oppr 20285  df-dvdsr 20308  df-rhm 20423  df-subrg 20520  df-lmod 20757  df-lss 20828  df-lsp 20868  df-sra 21070  df-rgmod 21071  df-lidl 21116  df-rsp 21117  df-2idl 21157

This theorem is referenced by:  aks5lem2  41790