Theorem rhmqusspan 42142

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 20273	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ Ring)
9		rhmqusspan.8	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))
10		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐺) = (Base‘𝐺)
11		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (RSpan‘𝐺) = (RSpan‘𝐺)
12		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (∥r‘𝐺) = (∥r‘𝐺)
13	10, 11, 12	rspsn 21366	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Ring ∧ 𝑋 ∈ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) = {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦})
14	8, 9, 13	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) = {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦})
15	14	eleq2d 2830	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ ((RSpan‘𝐺)‘{𝑋}) ↔ 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}))
16	15	biimpd 229	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ((RSpan‘𝐺)‘{𝑋}) → 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}))
17	16	imp 406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦})
18		vex 3492	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
19	18	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑥 ∈ V)
20		breq2 5170	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑋(∥r‘𝐺)𝑦 ↔ 𝑋(∥r‘𝐺)𝑥))
21	20	elabg 3690	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → (𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦} ↔ 𝑋(∥r‘𝐺)𝑥))
22	21	biimpd 229	. . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦}) → 𝑋(∥r‘𝐺)𝑥)
25		eqid 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (.r‘𝐺) = (.r‘𝐺)
26	10, 12, 25	dvdsr 20388	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋(∥r‘𝐺)𝑥 ↔ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥))
27	26	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋(∥r‘𝐺)𝑥 → (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥))
28	27	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑋(∥r‘𝐺)𝑥) → (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥))
29		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧(.r‘𝐺)𝑋) = 𝑥 → (𝐹‘(𝑧(.r‘𝐺)𝑋)) = (𝐹‘𝑥))
30	29	eqcomd 2746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧(.r‘𝐺)𝑋) = 𝑥 → (𝐹‘𝑥) = (𝐹‘(𝑧(.r‘𝐺)𝑋)))
31	30	adantl 481	. . . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
33		simpr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝑧 ∈ (Base‘𝐺))
34	9	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝑋 ∈ (Base‘𝐺))
35		eqid 2740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (.r‘𝐻) = (.r‘𝐻)
36	10, 25, 35	rhmmul 20512	. . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝐹‘𝑋) = 0 )
40	39	oveq2d 7464	. . . . . . . . . . . . . . . . . . . . . . 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 20503	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
42		ringsrg 20320	. . . . . . . . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Base‘𝐻) = (Base‘𝐻)
45	10, 44	rhmf 20511	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
48	47	ffvelcdmda 7118	. . . . . . . . . . . . . . . . . . . . . . . 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 20234	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐻 ∈ SRing ∧ (𝐹‘𝑧) ∈ (Base‘𝐻)) → ((𝐹‘𝑧)(.r‘𝐻) 0 ) = 0 )
50	43, 48, 49	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . 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 1913	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑧(𝑦(.r‘𝐺)𝑋) = 𝑥
56		nfv 1913	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑦(𝑧(.r‘𝐺)𝑋) = 𝑥
57		oveq1 7455	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 → (𝑦(.r‘𝐺)𝑋) = (𝑧(.r‘𝐺)𝑋))
58	57	eqeq1d 2742	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → ((𝑦(.r‘𝐺)𝑋) = 𝑥 ↔ (𝑧(.r‘𝐺)𝑋) = 𝑥))
59	55, 56, 58	cbvrexw 3313	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥 ↔ ∃𝑧 ∈ (Base‘𝐺)(𝑧(.r‘𝐺)𝑋) = 𝑥)
60	59	biimpi 216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥 → ∃𝑧 ∈ (Base‘𝐺)(𝑧(.r‘𝐺)𝑋) = 𝑥)
61	60	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥) → ∃𝑧 ∈ (Base‘𝐺)(𝑧(.r‘𝐺)𝑋) = 𝑥)
62	61	adantl 481	. . . . . . . . . . . . . . . . . . 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 3168	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) → (𝐹‘𝑥) = 0 )
64	63	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥) → (𝐹‘𝑥) = 0 ))
65	64	adantr 480	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋(∥r‘𝐺)𝑥 → (𝐹‘𝑥) = 0 ))
68	67	adantr 480	. . . . . . . . . . . . 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 412	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦} → (𝐹‘𝑥) = 0 ))
71	70	adantr 480	. . . . . . . . . 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 6935	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝐹‘𝑥) ∈ V)
74		elsng 4662	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝐹‘𝑥) ∈ { 0 })
77	46	ffund 6751	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
78	77	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → Fun 𝐹)
79		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (LIdeal‘𝐺) = (LIdeal‘𝐺)
80	79, 10	lidl1 21266	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Ring → (Base‘𝐺) ∈ (LIdeal‘𝐺))
81	8, 80	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐺) ∈ (LIdeal‘𝐺))
82	9	snssd 4834	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑋} ⊆ (Base‘𝐺))
83	11, 79	rspssp 21272	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Ring ∧ (Base‘𝐺) ∈ (LIdeal‘𝐺) ∧ {𝑋} ⊆ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) ⊆ (Base‘𝐺))
84	8, 81, 82, 83	syl3anc 1371	. . . . . . . . . . 11 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ (Base‘𝐺))
85	84	sselda 4008	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ (Base‘𝐺))
86		fdm 6756	. . . . . . . . . . . 12 ⊢ (𝐹:(Base‘𝐺)⟶(Base‘𝐻) → dom 𝐹 = (Base‘𝐺))
87	46, 86	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = (Base‘𝐺))
88	87	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → dom 𝐹 = (Base‘𝐺))
89	85, 88	eleqtrrd 2847	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ dom 𝐹)
90		fvimacnv 7086	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ { 0 } ↔ 𝑥 ∈ (◡𝐹 “ { 0 })))
91	78, 89, 90	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → ((𝐹‘𝑥) ∈ { 0 } ↔ 𝑥 ∈ (◡𝐹 “ { 0 })))
92	76, 91	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ (◡𝐹 “ { 0 }))
93	92	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((RSpan‘𝐺)‘{𝑋}) → 𝑥 ∈ (◡𝐹 “ { 0 })))
94	93	ssrdv 4014	. . . . 5 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ (◡𝐹 “ { 0 }))
95	3	eqcomi 2749	. . . . 5 ⊢ (◡𝐹 “ { 0 }) = 𝐾
96	94, 95	sseqtrdi 4059	. . . 4 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ 𝐾)
97	7, 96	eqsstrid 4057	. . 3 ⊢ (𝜑 → 𝑁 ⊆ 𝐾)
98	11, 10, 79	rspcl 21268	. . . . 5 ⊢ ((𝐺 ∈ Ring ∧ {𝑋} ⊆ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) ∈ (LIdeal‘𝐺))
99	8, 82, 98	syl2anc 583	. . . 4 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ∈ (LIdeal‘𝐺))
100	7, 99	eqeltrid 2848	. . 3 ⊢ (𝜑 → 𝑁 ∈ (LIdeal‘𝐺))
101	1, 2, 3, 4, 5, 6, 97, 100	rhmqusnsg 21318	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))
102	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
103		rhmghm 20510	. . . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝑁 ⊆ 𝐾)
106		lidlnsg 21281	. . . . . 6 ⊢ ((𝐺 ∈ Ring ∧ 𝑁 ∈ (LIdeal‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
107	8, 100, 106	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
108	107	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
109		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝑔 ∈ (Base‘𝐺))
110	1, 104, 3, 4, 5, 105, 108, 109	ghmqusnsglem1 19320	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → (𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔))
111	110	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑔 ∈ (Base‘𝐺)(𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔))
112	101, 111	jca 511	1 ⊢ (𝜑 → (𝐽 ∈ (𝑄 RingHom 𝐻) ∧ ∀𝑔 ∈ (Base‘𝐺)(𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  {csn 4648  ∪ cuni 4931   class class class wbr 5166   ↦ cmpt 5249  ^◡ccnv 5699  dom cdm 5700   “ cima 5703  Fun wfun 6567  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  [cec 8761  Basecbs 17258  ._rcmulr 17312  0_gc0g 17499   /_s cqus 17565  NrmSGrpcnsg 19161   ~_QG cqg 19162   GrpHom cghm 19252  SRingcsrg 20213  Ringcrg 20260  CRingccrg 20261  ∥_rcdsr 20380   RingHom crh 20495  LIdealclidl 21239  RSpancrsp 21240

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-int 4971  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-er 8763  df-ec 8765  df-qs 8769  df-map 8886  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-0g 17501  df-imas 17568  df-qus 17569  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  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-srg 20214  df-ring 20262  df-cring 20263  df-oppr 20360  df-dvdsr 20383  df-rhm 20498  df-subrg 20597  df-lmod 20882  df-lss 20953  df-lsp 20993  df-sra 21195  df-rgmod 21196  df-lidl 21241  df-rsp 21242  df-2idl 21283

This theorem is referenced by:  aks5lem2  42144