Theorem txlm 23546

Description: Two sequences converge iff the sequence of their ordered pairs converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)

Hypotheses

Ref	Expression
txlm.z	⊢ 𝑍 = (ℤ≥‘𝑀)
txlm.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
txlm.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
txlm.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
txlm.f	⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
txlm.g	⊢ (𝜑 → 𝐺:𝑍⟶𝑌)
txlm.h	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)

Assertion

Ref	Expression
txlm	⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩))

Distinct variable groups:   𝑛,𝐹   𝜑,𝑛   𝑛,𝐺   𝑛,𝐽   𝑛,𝐾   𝑛,𝑋   𝑛,𝑌   𝑛,𝑍

Allowed substitution hints:   𝑅(𝑛)   𝑆(𝑛)   𝐻(𝑛)   𝑀(𝑛)

Proof of Theorem txlm

Dummy variables 𝑗 𝑘 𝑢 𝑣 𝑤 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.27v 3183	. . . . . . . 8 ⊢ ((∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑢 ∈ 𝐽 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
2		r19.28v 3185	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
3	2	ralimi 3079	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝐽 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
4	1, 3	syl 17	. . . . . . 7 ⊢ ((∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
5		simprl 770	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → 𝑤 ∈ (𝐽 ×t 𝐾))
6		txlm.j	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
7		topontop 22809	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
8	6, 7	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐽 ∈ Top)
9		txlm.k	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
10		topontop 22809	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
11	9, 10	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ Top)
12		eqid 2728	. . . . . . . . . . . . . . . . . 18 ⊢ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))
13	12	txval 23462	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
14	8, 11, 13	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
15	14	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
16	5, 15	eleqtrd 2831	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → 𝑤 ∈ (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))))
17		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ⟨𝑅, 𝑆⟩ ∈ 𝑤)
18		tg2 22862	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ (topGen‘ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → ∃𝑡 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤))
19	16, 17, 18	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ∃𝑡 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤))
20		vex 3474	. . . . . . . . . . . . . . . 16 ⊢ 𝑢 ∈ V
21		vex 3474	. . . . . . . . . . . . . . . 16 ⊢ 𝑣 ∈ V
22	20, 21	xpex 7750	. . . . . . . . . . . . . . 15 ⊢ (𝑢 × 𝑣) ∈ V
23	22	rgen2w 3062	. . . . . . . . . . . . . 14 ⊢ ∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (𝑢 × 𝑣) ∈ V
24		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣)) = (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))
25		eleq2 2818	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑡 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣)))
26		sseq1 4004	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 × 𝑣) → (𝑡 ⊆ 𝑤 ↔ (𝑢 × 𝑣) ⊆ 𝑤))
27	25, 26	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑢 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
28	24, 27	rexrnmpo 7556	. . . . . . . . . . . . . 14 ⊢ (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 (𝑢 × 𝑣) ∈ V → (∃𝑡 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤) ↔ ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
29	23, 28	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (∃𝑡 ∈ ran (𝑢 ∈ 𝐽, 𝑣 ∈ 𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡 ∧ 𝑡 ⊆ 𝑤) ↔ ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
30	19, 29	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
31	30	ex 412	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
32		r19.29 3110	. . . . . . . . . . . . 13 ⊢ ((∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑢 ∈ 𝐽 (∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
33		r19.29 3110	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑣 ∈ 𝐾 (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
34		simprl 770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣))
35		opelxp 5709	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑣))
36	34, 35	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑣))
37		pm2.27 42	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ 𝑢 → ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))
38		pm2.27 42	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ 𝑣 → ((𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
39	37, 38	im2anan9 619	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑣) → (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
40	36, 39	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
41		txlm.z	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 = (ℤ≥‘𝑀)
42	41	rexanuz2 15323	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
43	41	uztrn2 12866	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
44		opelxpi 5710	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑣))
45		fveq2 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
46		fveq2 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
47	45, 46	opeq12d 4878	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 𝑘 → ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ = ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩)
48		txlm.h	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)
49		opex 5461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ V
50	47, 48, 49	fvmpt 7000	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ 𝑍 → (𝐻‘𝑘) = ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩)
51	50	eleq1d 2814	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ 𝑍 → ((𝐻‘𝑘) ∈ (𝑢 × 𝑣) ↔ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑣)))
52	44, 51	imbitrrid 245	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ 𝑍 → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ (𝑢 × 𝑣)))
53	52	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ (𝑢 × 𝑣)))
54		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (𝑢 × 𝑣) ⊆ 𝑤)
55	54	sseld 3978	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → ((𝐻‘𝑘) ∈ (𝑢 × 𝑣) → (𝐻‘𝑘) ∈ 𝑤))
56	53, 55	syld 47	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘 ∈ 𝑍) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ 𝑤))
57	43, 56	sylan2 592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗))) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ 𝑤))
58	57	anassrs 467	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → (𝐻‘𝑘) ∈ 𝑤))
59	58	ralimdva 3163	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
60	59	reximdva 3164	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ 𝑢 ∧ (𝐺‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
61	42, 60	biimtrrid 242	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ((∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
62	40, 61	syld 47	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
63	62	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
64	63	impcomd 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐾) → ((((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
65	64	rexlimdva 3151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐽) → (∃𝑣 ∈ 𝐾 (((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
66	33, 65	syl5 34	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐽) → ((∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
67	66	rexlimdva 3151	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑢 ∈ 𝐽 (∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
68	32, 67	syl5 34	. . . . . . . . . . . 12 ⊢ (𝜑 → ((∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ∧ ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))
69	68	expcomd 416	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
70	31, 69	syld 47	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
71	70	expdimp 452	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
72	71	com23 86	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
73	72	ralrimdva 3150	. . . . . . 7 ⊢ (𝜑 → (∀𝑢 ∈ 𝐽 ∀𝑣 ∈ 𝐾 ((𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
74	4, 73	syl5 34	. . . . . 6 ⊢ (𝜑 → ((∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
75	74	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → ((∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) → ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
76	8	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝐽 ∈ Top)
77	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝐾 ∈ Top)
78		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝑢 ∈ 𝐽)
79		toponmax 22822	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
80	9, 79	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
81	80	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝑌 ∈ 𝐾)
82		txopn 23500	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑢 ∈ 𝐽 ∧ 𝑌 ∈ 𝐾)) → (𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾))
83	76, 77, 78, 81, 82	syl22anc 838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾))
84		eleq2 2818	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑢 × 𝑌) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌)))
85		eleq2 2818	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑢 × 𝑌) → ((𝐻‘𝑘) ∈ 𝑤 ↔ (𝐻‘𝑘) ∈ (𝑢 × 𝑌)))
86	85	rexralbidv 3216	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑢 × 𝑌) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌)))
87	84, 86	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑢 × 𝑌) → ((⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌))))
88	87	rspcv 3604	. . . . . . . . . . 11 ⊢ ((𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌))))
89	83, 88	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌))))
90		simprl 770	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → 𝑆 ∈ 𝑌)
91		opelxpi 5710	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑢 ∧ 𝑆 ∈ 𝑌) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌))
92	90, 91	sylan2 592	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑢 ∧ (𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽))) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌))
93	92	expcom 413	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (𝑅 ∈ 𝑢 → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌)))
94	50	eleq1d 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝑍 → ((𝐻‘𝑘) ∈ (𝑢 × 𝑌) ↔ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑌)))
95		opelxp1 5715	. . . . . . . . . . . . . . . 16 ⊢ (⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑢 × 𝑌) → (𝐹‘𝑘) ∈ 𝑢)
96	94, 95	biimtrdi 252	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑍 → ((𝐻‘𝑘) ∈ (𝑢 × 𝑌) → (𝐹‘𝑘) ∈ 𝑢))
97	43, 96	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐻‘𝑘) ∈ (𝑢 × 𝑌) → (𝐹‘𝑘) ∈ 𝑢))
98	97	ralimdva 3163	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))
99	98	reximia 3077	. . . . . . . . . . . 12 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)
100	99	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))
101	93, 100	imim12d 81	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑢 × 𝑌)) → (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
102	89, 101	syld 47	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑆 ∈ 𝑌 ∧ 𝑢 ∈ 𝐽)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
103	102	anassrs 467	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑆 ∈ 𝑌) ∧ 𝑢 ∈ 𝐽) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
104	103	ralrimdva 3150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑌) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
105	104	adantrl 715	. . . . . 6 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)))
106	8	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝐽 ∈ Top)
107	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝐾 ∈ Top)
108		toponmax 22822	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
109	6, 108	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
110	109	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝑋 ∈ 𝐽)
111		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → 𝑣 ∈ 𝐾)
112		txopn 23500	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑋 ∈ 𝐽 ∧ 𝑣 ∈ 𝐾)) → (𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾))
113	106, 107, 110, 111, 112	syl22anc 838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾))
114		eleq2 2818	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑋 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
115		eleq2 2818	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑋 × 𝑣) → ((𝐻‘𝑘) ∈ 𝑤 ↔ (𝐻‘𝑘) ∈ (𝑋 × 𝑣)))
116	115	rexralbidv 3216	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑋 × 𝑣) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣)))
117	114, 116	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑋 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣))))
118	117	rspcv 3604	. . . . . . . . . . 11 ⊢ ((𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣))))
119	113, 118	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣))))
120		opelxpi 5710	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑣) → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣))
121	120	ex 412	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑋 → (𝑆 ∈ 𝑣 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
122	121	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (𝑆 ∈ 𝑣 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
123	50	eleq1d 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝑍 → ((𝐻‘𝑘) ∈ (𝑋 × 𝑣) ↔ ⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑋 × 𝑣)))
124		opelxp2 5716	. . . . . . . . . . . . . . . 16 ⊢ (⟨(𝐹‘𝑘), (𝐺‘𝑘)⟩ ∈ (𝑋 × 𝑣) → (𝐺‘𝑘) ∈ 𝑣)
125	123, 124	biimtrdi 252	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑍 → ((𝐻‘𝑘) ∈ (𝑋 × 𝑣) → (𝐺‘𝑘) ∈ 𝑣))
126	43, 125	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → ((𝐻‘𝑘) ∈ (𝑋 × 𝑣) → (𝐺‘𝑘) ∈ 𝑣))
127	126	ralimdva 3163	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
128	127	reximia 3077	. . . . . . . . . . . 12 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)
129	128	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))
130	122, 129	imim12d 81	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ (𝑋 × 𝑣)) → (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
131	119, 130	syld 47	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑣 ∈ 𝐾)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
132	131	anassrs 467	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝑋) ∧ 𝑣 ∈ 𝐾) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
133	132	ralrimdva 3150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝑋) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
134	133	adantrr 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))
135	105, 134	jcad 512	. . . . 5 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤) → (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))))
136	75, 135	impbid 211	. . . 4 ⊢ ((𝜑 ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌)) → ((∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)) ↔ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
137	136	pm5.32da 578	. . 3 ⊢ (𝜑 → (((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))))
138		opelxp 5709	. . . 4 ⊢ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ↔ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌))
139	138	anbi1i 623	. . 3 ⊢ ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤)))
140	137, 139	bitr4di 289	. 2 ⊢ (𝜑 → (((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))))
141		txlm.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
142		txlm.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
143		eqidd 2729	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
144	6, 41, 141, 142, 143	lmbrf 23158	. . . 4 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑅 ↔ (𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢))))
145		txlm.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)
146		eqidd 2729	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘𝑘))
147	9, 41, 141, 145, 146	lmbrf 23158	. . . 4 ⊢ (𝜑 → (𝐺(⇝𝑡‘𝐾)𝑆 ↔ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))))
148	144, 147	anbi12d 631	. . 3 ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ ((𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) ∧ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))))
149		an4 655	. . 3 ⊢ (((𝑅 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢)) ∧ (𝑆 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣))))
150	148, 149	bitrdi 287	. 2 ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ ((𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝑌) ∧ (∀𝑢 ∈ 𝐽 (𝑅 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ∈ 𝑢) ∧ ∀𝑣 ∈ 𝐾 (𝑆 ∈ 𝑣 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐺‘𝑘) ∈ 𝑣)))))
151		txtopon 23489	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
152	6, 9, 151	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
153	142	ffvelcdmda 7089	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ 𝑋)
154	145	ffvelcdmda 7089	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ 𝑌)
155	153, 154	opelxpd 5712	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩ ∈ (𝑋 × 𝑌))
156	155, 48	fmptd 7119	. . 3 ⊢ (𝜑 → 𝐻:𝑍⟶(𝑋 × 𝑌))
157		eqidd 2729	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐻‘𝑘))
158	152, 41, 141, 156, 157	lmbrf 23158	. 2 ⊢ (𝜑 → (𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩ ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐻‘𝑘) ∈ 𝑤))))
159	140, 150, 158	3bitr4d 311	1 ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3057  ∃wrex 3066  Vcvv 3470   ⊆ wss 3945  ⟨cop 4631   class class class wbr 5143   ↦ cmpt 5226   × cxp 5671  ran crn 5674  ⟶wf 6539  ‘cfv 6543  (class class class)co 7415   ∈ cmpo 7417  ℤcz 12583  ℤ_≥cuz 12847  topGenctg 17413  Topctop 22789  TopOnctopon 22806  ⇝_𝑡clm 23124   ×_t ctx 23458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-pre-lttri 11207  ax-pre-lttrn 11208

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-po 5585  df-so 5586  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7988  df-2nd 7989  df-er 8719  df-pm 8842  df-en 8959  df-dom 8960  df-sdom 8961  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-neg 11472  df-z 12584  df-uz 12848  df-topgen 17419  df-top 22790  df-topon 22807  df-bases 22843  df-lm 23127  df-tx 23460

This theorem is referenced by:  lmcn2  23547