Theorem 1arithufdlem4 33515

Description: Lemma for 1arithufd 33516. 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 2738	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 = (𝑀 Σg 𝑓) ↔ 𝑎 = (𝑀 Σg 𝑓)))
2	1	rexbidv 3166	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃𝑎 = (𝑀 Σg 𝑓)))
3		eqcom 2741	. . . . . . . . 9 ⊢ (𝑎 = (𝑀 Σg 𝑓) ↔ (𝑀 Σg 𝑓) = 𝑎)
4	3	rexbii 3082	. . . . . . . 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 33486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝐵)
12		oveq2 7421	. . . . . . . . 9 ⊢ (𝑓 = ⟨“𝑎”⟩ → (𝑀 Σg 𝑓) = (𝑀 Σg ⟨“𝑎”⟩))
13	12	eqeq1d 2736	. . . . . . . 8 ⊢ (𝑓 = ⟨“𝑎”⟩ → ((𝑀 Σg 𝑓) = 𝑎 ↔ (𝑀 Σg ⟨“𝑎”⟩) = 𝑎))
14	10	s1cld 14624	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ⟨“𝑎”⟩ ∈ Word 𝑃)
15		1arithufd.m	. . . . . . . . . . 11 ⊢ 𝑀 = (mulGrp‘𝑅)
16	15, 6	mgpbas 20111	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
17	16	gsumws1 18821	. . . . . . . . 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 3608	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = 𝑎)
20	5, 11, 19	elrabd 3677	. . . . . 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 2844	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑆)
23	22	ex 412	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝑃 → 𝑎 ∈ 𝑆))
24	23	ssrdv 3969	. . 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 4194	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑖 → (𝑆 ∩ 𝑝) = (𝑆 ∩ 𝑖))
28	27	eqeq1d 2736	. . . . . . . . . 10 ⊢ (𝑝 = 𝑖 → ((𝑆 ∩ 𝑝) = ∅ ↔ (𝑆 ∩ 𝑖) = ∅))
29		sseq2 3990	. . . . . . . . . 10 ⊢ (𝑝 = 𝑖 → (((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝 ↔ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖))
30	28, 29	anbi12d 632	. . . . . . . . 9 ⊢ (𝑝 = 𝑖 → (((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝) ↔ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)))
31	30	elrab 3675	. . . . . . . 8 ⊢ (𝑖 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑝)} ↔ (𝑖 ∈ (LIdeal‘𝑅) ∧ ((𝑆 ∩ 𝑖) = ∅ ∧ ((RSpan‘𝑅)‘{𝑋}) ⊆ 𝑖)))
32	31	anbi2i 623	. . . . . . 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 624	. . . . 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 4189	. . . . . . 7 ⊢ (𝑖 ∩ 𝑆) = (𝑆 ∩ 𝑖)
36		simpllr 775	. . . . . . . 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 2781	. . . . . 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 734	. . . . . . 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 768	. . . . . . . 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 33510	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ IDomn)
43	42	idomringd 20697	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
44		1arithufdlem.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
45		eqid 2734	. . . . . . . . . . . . 13 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
46	6, 45	rspsnid 33339	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ ((RSpan‘𝑅)‘{𝑋}))
47	43, 44, 46	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ((RSpan‘𝑅)‘{𝑋}))
48	47	ad5antr 734	. . . . . . . . . 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 3964	. . . . . . . . 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 4646	. . . . . . . . . . 11 ⊢ (𝑋 ≠ 0 → ¬ 𝑋 ∈ { 0 })
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ { 0 })
53	52	ad5antr 734	. . . . . . . . 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 3028	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑖 ∧ ¬ 𝑋 ∈ { 0 }) → 𝑖 ≠ { 0 })
55	49, 53, 54	syl2anc 584	. . . . . . . 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 4767	. . . . . . 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 4193	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝑗 ∩ 𝑃) = (𝑖 ∩ 𝑃))
58	57	neeq1d 2990	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝑖 ∩ 𝑃) ≠ ∅))
59		eqid 2734	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
60		1arithufd.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
61	59, 7, 60	isufd 33508	. . . . . . . . . 10 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑗 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅))
62	61	simprbi 496	. . . . . . . . 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 3606	. . . . . . 7 ⊢ ((𝑅 ∈ UFD ∧ 𝑖 ∈ ((PrmIdeal‘𝑅) ∖ {{ 0 }})) → (𝑖 ∩ 𝑃) ≠ ∅)
66	39, 56, 65	syl2anc 584	. . . . . 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 4383	. . . . . . . . 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 2944	. . . . . . 7 ⊢ ((𝑖 ∩ 𝑆) = ∅ → ((𝑖 ∩ 𝑃) ≠ ∅ → ¬ (𝑖 ∩ 𝑃) ⊆ (𝑖 ∩ 𝑆)))
70		sslin 4223	. . . . . . . 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 582	. . . 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 20696	. . . . 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 4789	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → {𝑋} ⊆ 𝐵)
81		eqid 2734	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
82	45, 6, 81	rspcl 21208	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → ((RSpan‘𝑅)‘{𝑋}) ∈ (LIdeal‘𝑅))
83	78, 80, 82	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → ((RSpan‘𝑅)‘{𝑋}) ∈ (LIdeal‘𝑅))
84	15	ringmgp 20205	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
85	43, 84	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Mnd)
86	21	ssrab3 4062	. . . . . . 7 ⊢ 𝑆 ⊆ 𝐵
87	86	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
88		eqeq1 2738	. . . . . . . . . 10 ⊢ (𝑥 = (1r‘𝑅) → (𝑥 = (𝑀 Σg 𝑓) ↔ (1r‘𝑅) = (𝑀 Σg 𝑓)))
89	88	rexbidv 3166	. . . . . . . . 9 ⊢ (𝑥 = (1r‘𝑅) → (∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓) ↔ ∃𝑓 ∈ Word 𝑃(1r‘𝑅) = (𝑀 Σg 𝑓)))
90		eqcom 2741	. . . . . . . . . 10 ⊢ ((1r‘𝑅) = (𝑀 Σg 𝑓) ↔ (𝑀 Σg 𝑓) = (1r‘𝑅))
91	90	rexbii 3082	. . . . . . . . 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 2734	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (1r‘𝑅)
94	6, 93	ringidcl 20231	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵)
95	43, 94	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵)
96		oveq2 7421	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑀 Σg 𝑓) = (𝑀 Σg ∅))
97	96	eqeq1d 2736	. . . . . . . . 9 ⊢ (𝑓 = ∅ → ((𝑀 Σg 𝑓) = (1r‘𝑅) ↔ (𝑀 Σg ∅) = (1r‘𝑅)))
98		wrd0 14560	. . . . . . . . . 10 ⊢ ∅ ∈ Word 𝑃
99	98	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ Word 𝑃)
100	15, 93	ringidval 20149	. . . . . . . . . . 11 ⊢ (1r‘𝑅) = (0g‘𝑀)
101	100	gsum0 18667	. . . . . . . . . 10 ⊢ (𝑀 Σg ∅) = (1r‘𝑅)
102	101	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 Σg ∅) = (1r‘𝑅))
103	97, 99, 102	rspcedvdw 3608	. . . . . . . 8 ⊢ (𝜑 → ∃𝑓 ∈ Word 𝑃(𝑀 Σg 𝑓) = (1r‘𝑅))
104	92, 95, 103	elrabd 3677	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)})
105	104, 21	eleqtrrdi 2844	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝑆)
106		1arithufd.u	. . . . . . . . 9 ⊢ 𝑈 = (Unit‘𝑅)
107	8	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → 𝑅 ∈ UFD)
108		1arithufdlem.2	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑅 ∈ DivRing)
109	108	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑆) ∧ 𝑏 ∈ 𝑆) → ¬ 𝑅 ∈ DivRing)
110		eqid 2734	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
111		simplr 768	. . . . . . . . 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 33513	. . . . . . . 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 3189	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎(.r‘𝑅)𝑏) ∈ 𝑆)
116	15, 110	mgpplusg 20110	. . . . . . . 8 ⊢ (.r‘𝑅) = (+g‘𝑀)
117	16, 100, 116	issubm 18786	. . . . . . 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 1373	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀))
120	119	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → 𝑆 ∈ (SubMnd‘𝑀))
121		neq0 4332	. . . . . . . . 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 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑅 ∈ UFD)
125	108	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → ¬ 𝑅 ∈ DivRing)
126	44	ad4antr 732	. . . . . . . . 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 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → ¬ 𝑋 ∈ 𝑈)
129	50	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑋 ≠ 0 )
130		simplr 768	. . . . . . . . 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 775	. . . . . . . . . . 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 4184	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑢 ∈ 𝑆)
134	131, 133	eqeltrrd 2834	. . . . . . . . 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 33514	. . . . . . . 8 ⊢ (((((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = (𝑦(.r‘𝑅)𝑋)) → 𝑋 ∈ 𝑆)
136	43	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑅 ∈ Ring)
137	44	ad2antrr 726	. . . . . . . . 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 4185	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑢 ∈ ((RSpan‘𝑅)‘{𝑋}))
140	6, 110, 45	elrspsn 21213	. . . . . . . . . 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 837	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → ∃𝑦 ∈ 𝐵 𝑢 = (𝑦(.r‘𝑅)𝑋))
143	135, 142	r19.29a 3149	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) ∧ 𝑢 ∈ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋}))) → 𝑋 ∈ 𝑆)
144	123, 143	exlimddv 1934	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → 𝑋 ∈ 𝑆)
145	144	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → 𝑋 ∈ 𝑆)
146		simplr 768	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) ∧ ¬ (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅) → ¬ 𝑋 ∈ 𝑆)
147	145, 146	condan 817	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → (𝑆 ∩ ((RSpan‘𝑅)‘{𝑋})) = ∅)
148		eqid 2734	. . . 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 33426	. . 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 3149	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝑆) → ¬ 𝑃 ⊆ 𝑆)
151	25, 150	condan 817	1 ⊢ (𝜑 → 𝑋 ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  {crab 3419   ∖ cdif 3928   ∩ cin 3930   ⊆ wss 3931   ⊊ wpss 3932  ∅c0 4313  {csn 4606  ‘cfv 6541  (class class class)co 7413  Word cword 14535  ⟨“cs1 14616  Basecbs 17230  ._rcmulr 17275  0_gc0g 17456   Σ_g cgsu 17457  Mndcmnd 18717  SubMndcsubmnd 18765  mulGrpcmgp 20106  1_rcur 20147  Ringcrg 20199  CRingccrg 20200  Unitcui 20324  RPrimecrpm 20401  IDomncidom 20662  DivRingcdr 20698  LIdealclidl 21179  RSpancrsp 21180  PrmIdealcprmidl 33403  UFDcufd 33506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-ac2 10485  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-rpss 7725  df-om 7870  df-1st 7996  df-2nd 7997  df-supp 8168  df-tpos 8233  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-oadd 8492  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-fsupp 9384  df-oi 9532  df-dju 9923  df-card 9961  df-ac 10138  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-n0 12510  df-xnn0 12583  df-z 12597  df-uz 12861  df-fz 13530  df-fzo 13677  df-seq 14025  df-hash 14353  df-word 14536  df-lsw 14584  df-concat 14592  df-s1 14617  df-substr 14662  df-pfx 14692  df-sets 17184  df-slot 17202  df-ndx 17214  df-base 17231  df-ress 17254  df-plusg 17287  df-mulr 17288  df-sca 17290  df-vsca 17291  df-ip 17292  df-0g 17458  df-gsum 17459  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-submnd 18767  df-grp 18924  df-minusg 18925  df-sbg 18926  df-subg 19111  df-cntz 19305  df-lsm 19623  df-cmn 19769  df-abl 19770  df-mgp 20107  df-rng 20119  df-ur 20148  df-ring 20201  df-cring 20202  df-oppr 20303  df-dvdsr 20326  df-unit 20327  df-invr 20357  df-rprm 20402  df-nzr 20482  df-subrg 20539  df-domn 20664  df-idom 20665  df-lmod 20829  df-lss 20899  df-lsp 20939  df-sra 21141  df-rgmod 21142  df-lidl 21181  df-rsp 21182  df-prmidl 33404  df-ufd 33507

This theorem is referenced by:  1arithufd  33516