Theorem 1arithufdlem4 33965

Description: Lemma for 1arithufd 33966. Nonzero ring, non-field case. Those trivial cases are handled in the final proof. (Contributed by Thierry Arnoux, 3-Jun-2025.)

Hypotheses

Ref	Expression
1arithufd.b	⊢ 𝐵 = (Base‘𝑅)
1arithufd.0	⊢ 0 = (0g‘𝑅)
1arithufd.u	⊢ 𝑈 = (Unit‘𝑅)
1arithufd.p	⊢ 𝑃 = (RPrime‘𝑅)
1arithufd.m	⊢ 𝑀 = (mulGrp‘𝑅)
1arithufd.r	⊢ (𝜑 → 𝑅 ∈ UFD)
1arithufdlem.2	⊢ (𝜑 → ¬ 𝑅 ∈ DivRing)
1arithufdlem.s	⊢ 𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)}
1arithufdlem.3	⊢ (𝜑 → 𝑋 ∈ 𝐵)
1arithufdlem.4	⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
1arithufdlem.5	⊢ (𝜑 → 𝑋 ≠ 0 )

Assertion

Ref	Expression
1arithufdlem4	⊢ (𝜑 → 𝑋 ∈ 𝑆)

Distinct variable groups:   0 ,𝑓   𝑥,𝐵   𝑓,𝑀,𝑥   𝑃,𝑓,𝑥   𝑅,𝑓   𝜑,𝑓,𝑥   𝑥,𝑋   𝑥,𝑈   𝑥,𝑅   𝑥,𝑆   𝑥, 0   𝑈,𝑓   𝐵,𝑓   𝑓,𝑋   𝑆,𝑓

Proof of Theorem 1arithufdlem4

Dummy variables 𝑝 𝑖 𝑗 𝑢 𝑎 𝑏 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 = (𝑀 Σg 𝑓) ↔ 𝑎 = (𝑀 Σg 𝑓)))
2	1	rexbidv 3188	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑎 = (𝑀 Σg 𝑓)))
3		eqcom 2769	. . . . . . . . 9 ⊢ (𝑎 = (𝑀 Σg 𝑓) ↔ (𝑀 Σg 𝑓) = 𝑎)
4	3	rexbii 3111	. . . . . . . 8 ⊢ (∃𝑓 ∈ Word 𝑃𝑎 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = 𝑎)
5	2, 4	bitrdi 290	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = 𝑎))
6		1arithufd.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
7		1arithufd.p	. . . . . . . 8 ⊢ 𝑃 = (RPrime‘𝑅)
8		1arithufd.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ UFD)
9	8	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑅 ∈ UFD)
10		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑃)
11	6, 7, 9, 10	rprmcl 33936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝐵)
12		oveq2 7425	. . . . . . . . 9 ⊢ (𝑓 = ⟨“𝑎”⟩ → (𝑀 Σg 𝑓) = (𝑀 Σg ⟨“𝑎”⟩))
13	12	eqeq1d 2764	. . . . . . . 8 ⊢ (𝑓 = ⟨“𝑎”⟩ → ((𝑀 Σg 𝑓) = 𝑎 ↔ (𝑀 Σg ⟨“𝑎”⟩) = 𝑎))
14	10	s1cld 14674	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ⟨“𝑎”⟩ ∈ Word 𝑃)
15		1arithufd.m	. . . . . . . . . . 11 ⊢ 𝑀 = (mulGrp‘𝑅)
16	15, 6	mgpbas 20284	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
17	16	gsumws1 18953	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐵 → (𝑀 Σg ⟨“𝑎”⟩) = 𝑎)
18	11, 17	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → (𝑀 Σg ⟨“𝑎”⟩) = 𝑎)
19	13, 14, 18	rspcedvdw 3582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = 𝑎)
20	5, 11, 19	elrabd 3650	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)})
21		1arithufdlem.s	. . . . . 6 ⊢ 𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)}
22	20, 21	eleqtrrdi 2873	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑆)
23	22	ex 418	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝑃 → 𝑎 ∈ 𝑆))
24	23	ssrdv 3940	. . 3 ⊢ (𝜑 → 𝑃 ⊆ 𝑆)
25	24	adantr 486	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑃 ⊆ 𝑆)
26		anass 474	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ↔ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ (𝑖 ∈ (LIdeal‘𝑅) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖))))
27		ineq2 4163	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑖 → (𝑆 ∩ 𝑝) = (𝑆 ∩ 𝑖))
28	27	eqeq1d 2764	. . . . . . . . . 10 ⊢ (𝑝 = 𝑖 → ((𝑆 ∩ 𝑝) = ∅ ↔ (𝑆 ∩ 𝑖) = ∅))
29		sseq2 3960	. . . . . . . . . 10 ⊢ (𝑝 = 𝑖 → (((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝 ↔ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖))
30	28, 29	anbi12d 644	. . . . . . . . 9 ⊢ (𝑝 = 𝑖 → (((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝) ↔ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)))
31	30	elrab 3648	. . . . . . . 8 ⊢ (𝑖 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ↔ (𝑖 ∈ (LIdeal‘𝑅) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)))
32	31	anbi2i 635	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)}) ↔ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ (𝑖 ∈ (LIdeal‘𝑅) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖))))
33	26, 32	bitr4i 281	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ↔ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)}))
34	33	anbi1i 636	. . . . 5 ⊢ (((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ↔ (((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)}) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)))
35		incom 4158	. . . . . . 7 ⊢ (𝑖 ∩ 𝑆) = (𝑆 ∩ 𝑖)
36		simpllr 788	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖))
37	36	simpld 500	. . . . . . 7 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → (𝑆 ∩ 𝑖) = ∅)
38	35, 37	eqtrid 2809	. . . . . 6 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → (𝑖 ∩ 𝑆) = ∅)
39	8	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → 𝑅 ∈ UFD)
40		simplr 781	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → 𝑖 ∈ (PrmIdeal‘𝑅))
41	36	simprd 501	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)
42	8	ufdidom 33960	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ IDomn)
43	42	idomringd 20895	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
44		1arithufdlem.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
45		eqid 2762	. . . . . . . . . . . . 13 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
46	6, 45	rspsnid 21442	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ ((RSpan‘𝑅)‘{𝑋}))
47	43, 44, 46	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ((RSpan‘𝑅)‘{𝑋}))
48	47	ad5antr 747	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → 𝑋 ∈ ((RSpan‘𝑅)‘{𝑋}))
49	41, 48	sseldd 3935	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → 𝑋 ∈ 𝑖)
50		1arithufdlem.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ 0 )
51		nelsn 4630	. . . . . . . . . . 11 ⊢ (𝑋 ≠ 0 → ¬ 𝑋 ∈ { 0 })
52	50, 51	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ { 0 })
53	52	ad5antr 747	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → ¬ 𝑋 ∈ { 0 })
54		nelne1 3054	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑖 ∧ ¬ 𝑋 ∈ { 0 }) → 𝑖 ≠ { 0 })
55	49, 53, 54	syl2anc 596	. . . . . . . 8 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → 𝑖 ≠ { 0 })
56	40, 55	eldifsnd 4753	. . . . . . 7 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → 𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }}))
57		ineq1 4162	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑗 ∩ 𝑃) = (𝑖 ∩ 𝑃))
58	57	neeq1d 3016	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝑖 ∩ 𝑃) ≠ ∅))
59		eqid 2762	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
60		1arithufd.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
61	59, 7, 60	isufd 33958	. . . . . . . . . 10 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑗 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅))
62	61	simprbi 503	. . . . . . . . 9 ⊢ (𝑅 ∈ UFD → ∀𝑗 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅)
63	62	adantr 486	. . . . . . . 8 ⊢ ((𝑅 ∈ UFD ∧ 𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})) → ∀𝑗 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅)
64		simpr 490	. . . . . . . 8 ⊢ ((𝑅 ∈ UFD ∧ 𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})) → 𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }}))
65	58, 63, 64	rspcdva 3580	. . . . . . 7 ⊢ ((𝑅 ∈ UFD ∧ 𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})) → (𝑖 ∩ 𝑃) ≠ ∅)
66	39, 56, 65	syl2anc 596	. . . . . 6 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → (𝑖 ∩ 𝑃) ≠ ∅)
67		sseq0 4357	. . . . . . . . 9 ⊢ (((𝑖 ∩ 𝑃) ⊆ (𝑖 ∩ 𝑆) ∧ (𝑖 ∩ 𝑆) = ∅) → (𝑖 ∩ 𝑃) = ∅)
68	67	expcom 419	. . . . . . . 8 ⊢ ((𝑖 ∩ 𝑆) = ∅ → ((𝑖 ∩ 𝑃) ⊆ (𝑖 ∩ 𝑆) → (𝑖 ∩ 𝑃) = ∅))
69	68	necon3ad 2970	. . . . . . 7 ⊢ ((𝑖 ∩ 𝑆) = ∅ → ((𝑖 ∩ 𝑃) ≠ ∅ → ¬ (𝑖 ∩ 𝑃) ⊆ (𝑖 ∩ 𝑆)))
70		sslin 4191	. . . . . . . 8 ⊢ (𝑃 ⊆ 𝑆 → (𝑖 ∩ 𝑃) ⊆ (𝑖 ∩ 𝑆))
71	70	con3i 155	. . . . . . 7 ⊢ (¬ (𝑖 ∩ 𝑃) ⊆ (𝑖 ∩ 𝑆) → ¬ 𝑃 ⊆ 𝑆)
72	69, 71	syl6 36	. . . . . 6 ⊢ ((𝑖 ∩ 𝑆) = ∅ → ((𝑖 ∩ 𝑃) ≠ ∅ → ¬ 𝑃 ⊆ 𝑆))
73	38, 66, 72	sylc 66	. . . . 5 ⊢ ((((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → ¬ 𝑃 ⊆ 𝑆)
74	34, 73	sylanbr 594	. . . 4 ⊢ (((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)}) ∧ 𝑖 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗) → ¬ 𝑃 ⊆ 𝑆)
75	74	anasss 472	. . 3 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑖 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)}) ∧ (𝑖 ∈ (PrmIdeal‘𝑅) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗)) → ¬ 𝑃 ⊆ 𝑆)
76	42	idomcringd 20894	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
77	76	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑅 ∈ CRing)
78	43	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑅 ∈ Ring)
79	44	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵)
80	79	snssd 4750	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → {𝑋} ⊆ 𝐵)
81		eqid 2762	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
82	45, 6, 81	rspcl 21433	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → ((RSpan‘𝑅)‘{𝑋}) ∈ (LIdeal‘𝑅))
83	78, 80, 82	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → ((RSpan‘𝑅)‘{𝑋}) ∈ (LIdeal‘𝑅))
84	15	ringmgp 20384	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
85	43, 84	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Mnd)
86	21	ssrab3 4033	. . . . . . 7 ⊢ 𝑆 ⊆ 𝐵
87	86	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
88		eqeq1 2766	. . . . . . . . . 10 ⊢ (𝑥 = (1r‘𝑅) → (𝑥 = (𝑀 Σg 𝑓) ↔ (1r‘𝑅) = (𝑀 Σg 𝑓)))
89	88	rexbidv 3188	. . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃(1r‘𝑅) = (𝑀 Σg 𝑓)))
90		eqcom 2769	. . . . . . . . . 10 ⊢ ((1r‘𝑅) = (𝑀 Σg 𝑓) ↔ (𝑀 Σg 𝑓) = (1r‘𝑅))
91	90	rexbii 3111	. . . . . . . . 9 ⊢ (∃𝑓 ∈ Word 𝑃(1r‘𝑅) = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = (1r‘𝑅))
92	89, 91	bitrdi 290	. . . . . . . 8 ⊢ (𝑥 = (1r‘𝑅) → (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = (1r‘𝑅)))
93		eqid 2762	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (1r‘𝑅)
94	6, 93	ringidcl 20412	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵)
95	43, 94	syl 18	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵)
96		oveq2 7425	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑀 Σg 𝑓) = (𝑀 Σg ∅))
97	96	eqeq1d 2764	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑀 Σg 𝑓) = (1r‘𝑅) ↔ (𝑀 Σg ∅) = (1r‘𝑅)))
98		wrd0 14608	. . . . . . . . . 10 ⊢ ∅ ∈ Word 𝑃
99	98	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ Word 𝑃)
100	15, 93	ringidval 20328	. . . . . . . . . . 11 ⊢ (1r‘𝑅) = (0g‘𝑀)
101	100	gsum0 18792	. . . . . . . . . 10 ⊢ (𝑀 Σg ∅) = (1r‘𝑅)
102	101	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 Σg ∅) = (1r‘𝑅))
103	97, 99, 102	rspcedvdw 3582	. . . . . . . 8 ⊢ (𝜑 → ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = (1r‘𝑅))
104	92, 95, 103	elrabd 3650	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)})
105	104, 21	eleqtrrdi 2873	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝑆)
106		1arithufd.u	. . . . . . . . 9 ⊢ 𝑈 = (Unit‘𝑅)
107	8	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → 𝑅 ∈ UFD)
108		1arithufdlem.2	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing)
109	108	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → ¬ 𝑅 ∈ DivRing)
110		eqid 2762	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
111		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → 𝑎 ∈ 𝑆)
112		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → 𝑏 ∈ 𝑆)
113	6, 60, 106, 7, 15, 107, 109, 21, 110, 111, 112	1arithufdlem2 33963	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → (𝑎(.r‘𝑅)𝑏) ∈ 𝑆)
114	113	anasss 472	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎(.r‘𝑅)𝑏) ∈ 𝑆)
115	114	ralrimivva 3207	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎(.r‘𝑅)𝑏) ∈ 𝑆)
116	15, 110	mgpplusg 20283	. . . . . . . 8 ⊢ (.r‘𝑅) = (+g‘𝑀)
117	16, 100, 116	issubm 18917	. . . . . . 7 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝑆 ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎(.r‘𝑅)𝑏) ∈ 𝑆)))
118	117	biimpar 483	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ (𝑆 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝑆 ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎(.r‘𝑅)𝑏) ∈ 𝑆)) → 𝑆 ∈ (SubMnd‘𝑀))
119	85, 87, 105, 115, 118	syl13anc 1399	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀))
120	119	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑆 ∈ (SubMnd‘𝑀))
121		neq0 4302	. . . . . . . 8 ⊢ (¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅ ↔ ∃𝑢 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})))
122	121	bilani 510	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → ∃𝑢 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})))
123	8	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑅 ∈ UFD)
124	108	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → ¬ 𝑅 ∈ DivRing)
125	44	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑋 ∈ 𝐵)
126		1arithufdlem.4	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
127	126	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → ¬ 𝑋 ∈ 𝑈)
128	50	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑋 ≠ 0 )
129		simplr 781	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑦 ∈ 𝐵)
130		simpr 490	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑢 = (𝑦(.r‘𝑅)𝑋))
131		simpllr 788	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})))
132	131	elin1d 4153	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑢 ∈ 𝑆)
133	130, 132	eqeltrrd 2863	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → (𝑦(.r‘𝑅)𝑋) ∈ 𝑆)
134	6, 60, 106, 7, 15, 123, 124, 21, 125, 127, 128, 110, 129, 133	1arithufdlem3 33964	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑋 ∈ 𝑆)
135	43	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑅 ∈ Ring)
136	44	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑋 ∈ 𝐵)
137		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})))
138	137	elin2d 4154	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑢 ∈ ((RSpan‘𝑅)‘{𝑋}))
139	6, 110, 45	elrspsn 21440	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑢 ∈ ((RSpan‘𝑅)‘{𝑋}) ↔ ∃𝑦 ∈ 𝐵 𝑢 = (𝑦(.r‘𝑅)𝑋)))
140	139	biimpa 482	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑢 ∈ ((RSpan‘𝑅)‘{𝑋})) → ∃𝑦 ∈ 𝐵 𝑢 = (𝑦(.r‘𝑅)𝑋))
141	135, 136, 138, 140	syl21anc 851	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → ∃𝑦 ∈ 𝐵 𝑢 = (𝑦(.r‘𝑅)𝑋))
142	134, 141	r19.29a 3172	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑋 ∈ 𝑆)
143	122, 142	exlimddv 1968	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → 𝑋 ∈ 𝑆)
144	143	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → 𝑋 ∈ 𝑆)
145		simplr 781	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → ¬ 𝑋 ∈ 𝑆)
146	144, 145	condan 830	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅)
147		eqid 2762	. . . 4 ⊢ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)}
148	6, 77, 83, 120, 15, 146, 147	ssdifidlprm 21555	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → ∃𝑖 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} (𝑖 ∈ (PrmIdeal‘𝑅) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ¬ 𝑖 ⊊ 𝑗))
149	75, 148	r19.29a 3172	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → ¬ 𝑃 ⊆ 𝑆)
150	25, 149	condan 830	1 ⊢ (𝜑 → 𝑋 ∈ 𝑆)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4282  {csn 4587  ‘cfv 6537  (class class class)co 7417  Word cword 14582  ⟨“cs1 14666  Basecbs 17307  ._rcmulr 17349  0_gc0g 17530   Σ_g cgsu 17531  Mndcmnd 18842  SubMndcsubmnd 18896  mulGrpcmgp 20279  1_rcur 20326  Ringcrg 20378  CRingccrg 20379  Unitcui 20502  RPrimecrpm 20579  IDomncidom 20861  DivRingcdr 20896  LIdealclidl 21399  RSpancrsp 21400  PrmIdealcprmidl 21529  UFDcufd 33956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-ac2 10469  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-rpss 7728  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8163  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-er 8700  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-oi 9486  df-dju 9910  df-card 9948  df-ac 10123  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-n0 12533  df-xnn0 12606  df-z 12620  df-uz 12892  df-fz 13566  df-fzo 13714  df-seq 14070  df-hash 14399  df-word 14583  df-lsw 14632  df-concat 14640  df-s1 14667  df-substr 14713  df-pfx 14745  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-sca 17364  df-vsca 17365  df-ip 17366  df-0g 17532  df-gsum 17533  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-submnd 18898  df-grp 19066  df-minusg 19067  df-sbg 19068  df-subg 19252  df-cntz 19450  df-lsm 19769  df-cmn 19915  df-abl 19916  df-mgp 20280  df-rng 20294  df-ur 20327  df-ring 20380  df-cring 20381  df-oppr 20484  df-dvdsr 20504  df-unit 20505  df-invr 20535  df-rprm 20580  df-nzr 20679  df-subrg 20738  df-domn 20863  df-idom 20864  df-lmod 21052  df-lss 21122  df-lsp 21162  df-sra 21363  df-rgmod 21364  df-lidl 21401  df-rsp 21402  df-prmidl 21530  df-ufd 33957

This theorem is used by:  1arithufd  33966