Theorem 1arithufdlem4 33639

Description: Lemma for 1arithufd 33640. 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 2741	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 = (𝑀 Σg 𝑓) ↔ 𝑎 = (𝑀 Σg 𝑓)))
2	1	rexbidv 3162	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑎 = (𝑀 Σg 𝑓)))
3		eqcom 2744	. . . . . . . . 9 ⊢ (𝑎 = (𝑀 Σg 𝑓) ↔ (𝑀 Σg 𝑓) = 𝑎)
4	3	rexbii 3085	. . . . . . . 8 ⊢ (∃𝑓 ∈ Word 𝑃𝑎 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = 𝑎)
5	2, 4	bitrdi 287	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑅 ∈ UFD)
10		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑃)
11	6, 7, 9, 10	rprmcl 33610	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝐵)
12		oveq2 7376	. . . . . . . . 9 ⊢ (𝑓 = ⟨“𝑎”⟩ → (𝑀 Σg 𝑓) = (𝑀 Σg ⟨“𝑎”⟩))
13	12	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑓 = ⟨“𝑎”⟩ → ((𝑀 Σg 𝑓) = 𝑎 ↔ (𝑀 Σg ⟨“𝑎”⟩) = 𝑎))
14	10	s1cld 14539	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ⟨“𝑎”⟩ ∈ Word 𝑃)
15		1arithufd.m	. . . . . . . . . . 11 ⊢ 𝑀 = (mulGrp‘𝑅)
16	15, 6	mgpbas 20092	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
17	16	gsumws1 18775	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐵 → (𝑀 Σg ⟨“𝑎”⟩) = 𝑎)
18	11, 17	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → (𝑀 Σg ⟨“𝑎”⟩) = 𝑎)
19	13, 14, 18	rspcedvdw 3581	. . . . . . 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 2848	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑆)
23	22	ex 412	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝑃 → 𝑎 ∈ 𝑆))
24	23	ssrdv 3941	. . 3 ⊢ (𝜑 → 𝑃 ⊆ 𝑆)
25	24	adantr 480	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑃 ⊆ 𝑆)
26		anass 468	. . . . . . 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 4168	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑖 → (𝑆 ∩ 𝑝) = (𝑆 ∩ 𝑖))
28	27	eqeq1d 2739	. . . . . . . . . 10 ⊢ (𝑝 = 𝑖 → ((𝑆 ∩ 𝑝) = ∅ ↔ (𝑆 ∩ 𝑖) = ∅))
29		sseq2 3962	. . . . . . . . . 10 ⊢ (𝑝 = 𝑖 → (((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝 ↔ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖))
30	28, 29	anbi12d 633	. . . . . . . . 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 624	. . . . . . 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 278	. . . . . 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 625	. . . . 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 4163	. . . . . . 7 ⊢ (𝑖 ∩ 𝑆) = (𝑆 ∩ 𝑖)
36		simpllr 776	. . . . . . . 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 494	. . . . . . 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 2784	. . . . . 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 735	. . . . . . 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 769	. . . . . . . 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 495	. . . . . . . . . 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 33634	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ IDomn)
43	42	idomringd 20673	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
44		1arithufdlem.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
45		eqid 2737	. . . . . . . . . . . . 13 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
46	6, 45	rspsnid 33463	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ ((RSpan‘𝑅)‘{𝑋}))
47	43, 44, 46	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ((RSpan‘𝑅)‘{𝑋}))
48	47	ad5antr 735	. . . . . . . . . 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 3936	. . . . . . . . 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 4625	. . . . . . . . . . 11 ⊢ (𝑋 ≠ 0 → ¬ 𝑋 ∈ { 0 })
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ { 0 })
53	52	ad5antr 735	. . . . . . . . 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 3030	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑖 ∧ ¬ 𝑋 ∈ { 0 }) → 𝑖 ≠ { 0 })
55	49, 53, 54	syl2anc 585	. . . . . . . 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 4745	. . . . . . 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 4167	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑗 ∩ 𝑃) = (𝑖 ∩ 𝑃))
58	57	neeq1d 2992	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝑖 ∩ 𝑃) ≠ ∅))
59		eqid 2737	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
60		1arithufd.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
61	59, 7, 60	isufd 33632	. . . . . . . . . 10 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑗 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅))
62	61	simprbi 497	. . . . . . . . 9 ⊢ (𝑅 ∈ UFD → ∀𝑗 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅)
63	62	adantr 480	. . . . . . . 8 ⊢ ((𝑅 ∈ UFD ∧ 𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})) → ∀𝑗 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅)
64		simpr 484	. . . . . . . 8 ⊢ ((𝑅 ∈ UFD ∧ 𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})) → 𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }}))
65	58, 63, 64	rspcdva 3579	. . . . . . 7 ⊢ ((𝑅 ∈ UFD ∧ 𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})) → (𝑖 ∩ 𝑃) ≠ ∅)
66	39, 56, 65	syl2anc 585	. . . . . 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 413	. . . . . . . 8 ⊢ ((𝑖 ∩ 𝑆) = ∅ → ((𝑖 ∩ 𝑃) ⊆ (𝑖 ∩ 𝑆) → (𝑖 ∩ 𝑃) = ∅))
69	68	necon3ad 2946	. . . . . . 7 ⊢ ((𝑖 ∩ 𝑆) = ∅ → ((𝑖 ∩ 𝑃) ≠ ∅ → ¬ (𝑖 ∩ 𝑃) ⊆ (𝑖 ∩ 𝑆)))
70		sslin 4197	. . . . . . . 8 ⊢ (𝑃 ⊆ 𝑆 → (𝑖 ∩ 𝑃) ⊆ (𝑖 ∩ 𝑆))
71	70	con3i 154	. . . . . . 7 ⊢ (¬ (𝑖 ∩ 𝑃) ⊆ (𝑖 ∩ 𝑆) → ¬ 𝑃 ⊆ 𝑆)
72	69, 71	syl6 35	. . . . . 6 ⊢ ((𝑖 ∩ 𝑆) = ∅ → ((𝑖 ∩ 𝑃) ≠ ∅ → ¬ 𝑃 ⊆ 𝑆))
73	38, 66, 72	sylc 65	. . . . 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 583	. . . 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 466	. . 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 20672	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
77	76	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑅 ∈ CRing)
78	43	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑅 ∈ Ring)
79	44	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵)
80	79	snssd 4767	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → {𝑋} ⊆ 𝐵)
81		eqid 2737	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
82	45, 6, 81	rspcl 21202	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → ((RSpan‘𝑅)‘{𝑋}) ∈ (LIdeal‘𝑅))
83	78, 80, 82	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → ((RSpan‘𝑅)‘{𝑋}) ∈ (LIdeal‘𝑅))
84	15	ringmgp 20186	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
85	43, 84	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Mnd)
86	21	ssrab3 4036	. . . . . . 7 ⊢ 𝑆 ⊆ 𝐵
87	86	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
88		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑥 = (1r‘𝑅) → (𝑥 = (𝑀 Σg 𝑓) ↔ (1r‘𝑅) = (𝑀 Σg 𝑓)))
89	88	rexbidv 3162	. . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃(1r‘𝑅) = (𝑀 Σg 𝑓)))
90		eqcom 2744	. . . . . . . . . 10 ⊢ ((1r‘𝑅) = (𝑀 Σg 𝑓) ↔ (𝑀 Σg 𝑓) = (1r‘𝑅))
91	90	rexbii 3085	. . . . . . . . 9 ⊢ (∃𝑓 ∈ Word 𝑃(1r‘𝑅) = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = (1r‘𝑅))
92	89, 91	bitrdi 287	. . . . . . . 8 ⊢ (𝑥 = (1r‘𝑅) → (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = (1r‘𝑅)))
93		eqid 2737	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (1r‘𝑅)
94	6, 93	ringidcl 20212	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵)
95	43, 94	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵)
96		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑀 Σg 𝑓) = (𝑀 Σg ∅))
97	96	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑀 Σg 𝑓) = (1r‘𝑅) ↔ (𝑀 Σg ∅) = (1r‘𝑅)))
98		wrd0 14474	. . . . . . . . . 10 ⊢ ∅ ∈ Word 𝑃
99	98	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ Word 𝑃)
100	15, 93	ringidval 20130	. . . . . . . . . . 11 ⊢ (1r‘𝑅) = (0g‘𝑀)
101	100	gsum0 18621	. . . . . . . . . 10 ⊢ (𝑀 Σg ∅) = (1r‘𝑅)
102	101	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 Σg ∅) = (1r‘𝑅))
103	97, 99, 102	rspcedvdw 3581	. . . . . . . 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 2848	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝑆)
106		1arithufd.u	. . . . . . . . 9 ⊢ 𝑈 = (Unit‘𝑅)
107	8	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → 𝑅 ∈ UFD)
108		1arithufdlem.2	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing)
109	108	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → ¬ 𝑅 ∈ DivRing)
110		eqid 2737	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
111		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → 𝑎 ∈ 𝑆)
112		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → 𝑏 ∈ 𝑆)
113	6, 60, 106, 7, 15, 107, 109, 21, 110, 111, 112	1arithufdlem2 33637	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → (𝑎(.r‘𝑅)𝑏) ∈ 𝑆)
114	113	anasss 466	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎(.r‘𝑅)𝑏) ∈ 𝑆)
115	114	ralrimivva 3181	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎(.r‘𝑅)𝑏) ∈ 𝑆)
116	15, 110	mgpplusg 20091	. . . . . . . 8 ⊢ (.r‘𝑅) = (+g‘𝑀)
117	16, 100, 116	issubm 18740	. . . . . . 7 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ (1r‘𝑅) ∈ 𝑆 ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎(.r‘𝑅)𝑏) ∈ 𝑆)))
118	117	biimpar 477	. . . . . 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 1375	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀))
120	119	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑆 ∈ (SubMnd‘𝑀))
121		neq0 4306	. . . . . . . . 9 ⊢ (¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅ ↔ ∃𝑢 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})))
122	121	biimpi 216	. . . . . . . 8 ⊢ (¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅ → ∃𝑢 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})))
123	122	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → ∃𝑢 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})))
124	8	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑅 ∈ UFD)
125	108	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → ¬ 𝑅 ∈ DivRing)
126	44	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑋 ∈ 𝐵)
127		1arithufdlem.4	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
128	127	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → ¬ 𝑋 ∈ 𝑈)
129	50	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑋 ≠ 0 )
130		simplr 769	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑦 ∈ 𝐵)
131		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑢 = (𝑦(.r‘𝑅)𝑋))
132		simpllr 776	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})))
133	132	elin1d 4158	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑢 ∈ 𝑆)
134	131, 133	eqeltrrd 2838	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → (𝑦(.r‘𝑅)𝑋) ∈ 𝑆)
135	6, 60, 106, 7, 15, 124, 125, 21, 126, 128, 129, 110, 130, 134	1arithufdlem3 33638	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑋 ∈ 𝑆)
136	43	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑅 ∈ Ring)
137	44	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑋 ∈ 𝐵)
138		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})))
139	138	elin2d 4159	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑢 ∈ ((RSpan‘𝑅)‘{𝑋}))
140	6, 110, 45	elrspsn 21207	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑢 ∈ ((RSpan‘𝑅)‘{𝑋}) ↔ ∃𝑦 ∈ 𝐵 𝑢 = (𝑦(.r‘𝑅)𝑋)))
141	140	biimpa 476	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑢 ∈ ((RSpan‘𝑅)‘{𝑋})) → ∃𝑦 ∈ 𝐵 𝑢 = (𝑦(.r‘𝑅)𝑋))
142	136, 137, 139, 141	syl21anc 838	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → ∃𝑦 ∈ 𝐵 𝑢 = (𝑦(.r‘𝑅)𝑋))
143	135, 142	r19.29a 3146	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑋 ∈ 𝑆)
144	123, 143	exlimddv 1937	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → 𝑋 ∈ 𝑆)
145	144	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → 𝑋 ∈ 𝑆)
146		simplr 769	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → ¬ 𝑋 ∈ 𝑆)
147	145, 146	condan 818	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅)
148		eqid 2737	. . . 4 ⊢ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)}
149	6, 77, 83, 120, 15, 147, 148	ssdifidlprm 33550	. . 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))
150	75, 149	r19.29a 3146	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → ¬ 𝑃 ⊆ 𝑆)
151	25, 150	condan 818	1 ⊢ (𝜑 → 𝑋 ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903   ⊊ wpss 3904  ∅c0 4287  {csn 4582  ‘cfv 6500  (class class class)co 7368  Word cword 14448  ⟨“cs1 14531  Basecbs 17148  ._rcmulr 17190  0_gc0g 17371   Σ_g cgsu 17372  Mndcmnd 18671  SubMndcsubmnd 18719  mulGrpcmgp 20087  1_rcur 20128  Ringcrg 20180  CRingccrg 20181  Unitcui 20303  RPrimecrpm 20380  IDomncidom 20638  DivRingcdr 20674  LIdealclidl 21173  RSpancrsp 21174  PrmIdealcprmidl 33527  UFDcufd 33630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-rpss 7678  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-tpos 8178  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-oi 9427  df-dju 9825  df-card 9863  df-ac 10038  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-n0 12414  df-xnn0 12487  df-z 12501  df-uz 12764  df-fz 13436  df-fzo 13583  df-seq 13937  df-hash 14266  df-word 14449  df-lsw 14498  df-concat 14506  df-s1 14532  df-substr 14577  df-pfx 14607  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-sca 17205  df-vsca 17206  df-ip 17207  df-0g 17373  df-gsum 17374  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-submnd 18721  df-grp 18878  df-minusg 18879  df-sbg 18880  df-subg 19065  df-cntz 19258  df-lsm 19577  df-cmn 19723  df-abl 19724  df-mgp 20088  df-rng 20100  df-ur 20129  df-ring 20182  df-cring 20183  df-oppr 20285  df-dvdsr 20305  df-unit 20306  df-invr 20336  df-rprm 20381  df-nzr 20458  df-subrg 20515  df-domn 20640  df-idom 20641  df-lmod 20825  df-lss 20895  df-lsp 20935  df-sra 21137  df-rgmod 21138  df-lidl 21175  df-rsp 21176  df-prmidl 33528  df-ufd 33631

This theorem is referenced by:  1arithufd  33640