Theorem rhmqusspan 42162

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 20131	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ Ring)
9		rhmqusspan.8	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))
10		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐺) = (Base‘𝐺)
11		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (RSpan‘𝐺) = (RSpan‘𝐺)
12		eqid 2729	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) = {𝑦 ∣ 𝑋(∥r‘𝐺)𝑦})
15	14	eleq2d 2814	. . . . . . . . . . . 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 3440	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
19	18	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑥 ∈ V)
20		breq2 5096	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑋(∥r‘𝐺)𝑦 ↔ 𝑋(∥r‘𝐺)𝑥))
21	20	elabg 3632	. . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (.r‘𝐺) = (.r‘𝐺)
26	10, 12, 25	dvdsr 20247	. . . . . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧(.r‘𝐺)𝑋) = 𝑥 → (𝐹‘(𝑧(.r‘𝐺)𝑋)) = (𝐹‘𝑥))
30	29	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → 𝑋 ∈ (Base‘𝐺))
35		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (.r‘𝐻) = (.r‘𝐻)
36	10, 25, 35	rhmmul 20371	. . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐺) ∧ ∃𝑦 ∈ (Base‘𝐺)(𝑦(.r‘𝐺)𝑋) = 𝑥)) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝐹‘𝑋) = 0 )
40	39	oveq2d 7365	. . . . . . . . . . . . . . . . . . . . . . 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 20362	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
42		ringsrg 20182	. . . . . . . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Base‘𝐻) = (Base‘𝐻)
45	10, 44	rhmf 20370	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7018	. . . . . . . . . . . . . . . . . . . . . . . 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 20092	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐻 ∈ SRing ∧ (𝐹‘𝑧) ∈ (Base‘𝐻)) → ((𝐹‘𝑧)(.r‘𝐻) 0 ) = 0 )
50	43, 48, 49	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . 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 1914	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑧(𝑦(.r‘𝐺)𝑋) = 𝑥
56		nfv 1914	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑦(𝑧(.r‘𝐺)𝑋) = 𝑥
57		oveq1 7356	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 → (𝑦(.r‘𝐺)𝑋) = (𝑧(.r‘𝐺)𝑋))
58	57	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → ((𝑦(.r‘𝐺)𝑋) = 𝑥 ↔ (𝑧(.r‘𝐺)𝑋) = 𝑥))
59	55, 56, 58	cbvrexw 3272	. . . . . . . . . . . . . . . . . . . . . 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 3137	. . . . . . . . . . . . . . . . . 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 6837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → (𝐹‘𝑥) ∈ V)
74		elsng 4591	. . . . . . . . . 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 6656	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
78	77	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → Fun 𝐹)
79		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (LIdeal‘𝐺) = (LIdeal‘𝐺)
80	79, 10	lidl1 21140	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Ring → (Base‘𝐺) ∈ (LIdeal‘𝐺))
81	8, 80	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐺) ∈ (LIdeal‘𝐺))
82	9	snssd 4760	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑋} ⊆ (Base‘𝐺))
83	11, 79	rspssp 21146	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Ring ∧ (Base‘𝐺) ∈ (LIdeal‘𝐺) ∧ {𝑋} ⊆ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) ⊆ (Base‘𝐺))
84	8, 81, 82, 83	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ (Base‘𝐺))
85	84	sselda 3935	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ (Base‘𝐺))
86		fdm 6661	. . . . . . . . . . . 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 2831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((RSpan‘𝐺)‘{𝑋})) → 𝑥 ∈ dom 𝐹)
90		fvimacnv 6987	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ { 0 } ↔ 𝑥 ∈ (◡𝐹 “ { 0 })))
91	78, 89, 90	syl2anc 584	. . . . . . . 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 3941	. . . . 5 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ (◡𝐹 “ { 0 }))
95	3	eqcomi 2738	. . . . 5 ⊢ (◡𝐹 “ { 0 }) = 𝐾
96	94, 95	sseqtrdi 3976	. . . 4 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ⊆ 𝐾)
97	7, 96	eqsstrid 3974	. . 3 ⊢ (𝜑 → 𝑁 ⊆ 𝐾)
98	11, 10, 79	rspcl 21142	. . . . 5 ⊢ ((𝐺 ∈ Ring ∧ {𝑋} ⊆ (Base‘𝐺)) → ((RSpan‘𝐺)‘{𝑋}) ∈ (LIdeal‘𝐺))
99	8, 82, 98	syl2anc 584	. . . 4 ⊢ (𝜑 → ((RSpan‘𝐺)‘{𝑋}) ∈ (LIdeal‘𝐺))
100	7, 99	eqeltrid 2832	. . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
103		rhmghm 20369	. . . . 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 21155	. . . . . 6 ⊢ ((𝐺 ∈ Ring ∧ 𝑁 ∈ (LIdeal‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
107	8, 100, 106	syl2anc 584	. . . . 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 19159	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (Base‘𝐺)) → (𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔))
111	110	ralrimiva 3121	. 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 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ⊆ wss 3903  {csn 4577  ∪ cuni 4858   class class class wbr 5092   ↦ cmpt 5173  ^◡ccnv 5618  dom cdm 5619   “ cima 5622  Fun wfun 6476  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  [cec 8623  Basecbs 17120  ._rcmulr 17162  0_gc0g 17343   /_s cqus 17409  NrmSGrpcnsg 19000   ~_QG cqg 19001   GrpHom cghm 19091  SRingcsrg 20071  Ringcrg 20118  CRingccrg 20119  ∥_rcdsr 20239   RingHom crh 20354  LIdealclidl 21113  RSpancrsp 21114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  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 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-tpos 8159  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-ec 8627  df-qs 8631  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-0g 17345  df-imas 17412  df-qus 17413  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-grp 18815  df-minusg 18816  df-sbg 18817  df-subg 19002  df-nsg 19003  df-eqg 19004  df-ghm 19092  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-srg 20072  df-ring 20120  df-cring 20121  df-oppr 20222  df-dvdsr 20242  df-rhm 20357  df-subrg 20455  df-lmod 20765  df-lss 20835  df-lsp 20875  df-sra 21077  df-rgmod 21078  df-lidl 21115  df-rsp 21116  df-2idl 21157

This theorem is referenced by:  aks5lem2  42164