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Theorem txlm 22184
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 = (ℤ𝑀)
txlm.m (𝜑𝑀 ∈ ℤ)
txlm.j (𝜑𝐽 ∈ (TopOn‘𝑋))
txlm.k (𝜑𝐾 ∈ (TopOn‘𝑌))
txlm.f (𝜑𝐹:𝑍𝑋)
txlm.g (𝜑𝐺:𝑍𝑌)
txlm.h 𝐻 = (𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)
Assertion
Ref Expression
txlm (𝜑 → ((𝐹(⇝𝑡𝐽)𝑅𝐺(⇝𝑡𝐾)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.
StepHypRef Expression
1 r19.27v 3181 . . . . . . . 8 ((∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∀𝑢𝐽 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
2 r19.28v 3183 . . . . . . . . 9 (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∀𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
32ralimi 3157 . . . . . . . 8 (∀𝑢𝐽 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
41, 3syl 17 . . . . . . 7 ((∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
5 simprl 767 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → 𝑤 ∈ (𝐽 ×t 𝐾))
6 txlm.j . . . . . . . . . . . . . . . . . 18 (𝜑𝐽 ∈ (TopOn‘𝑋))
7 topontop 21449 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
86, 7syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐽 ∈ Top)
9 txlm.k . . . . . . . . . . . . . . . . . 18 (𝜑𝐾 ∈ (TopOn‘𝑌))
10 topontop 21449 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
119, 10syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ Top)
12 eqid 2818 . . . . . . . . . . . . . . . . . 18 ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)) = ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))
1312txval 22100 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
148, 11, 13syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
1514adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
165, 15eleqtrd 2912 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → 𝑤 ∈ (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
17 simprr 769 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ⟨𝑅, 𝑆⟩ ∈ 𝑤)
18 tg2 21501 . . . . . . . . . . . . . 14 ((𝑤 ∈ (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → ∃𝑡 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡𝑡𝑤))
1916, 17, 18syl2anc 584 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ∃𝑡 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡𝑡𝑤))
20 vex 3495 . . . . . . . . . . . . . . . 16 𝑢 ∈ V
21 vex 3495 . . . . . . . . . . . . . . . 16 𝑣 ∈ V
2220, 21xpex 7465 . . . . . . . . . . . . . . 15 (𝑢 × 𝑣) ∈ V
2322rgen2w 3148 . . . . . . . . . . . . . 14 𝑢𝐽𝑣𝐾 (𝑢 × 𝑣) ∈ V
24 eqid 2818 . . . . . . . . . . . . . . 15 (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)) = (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))
25 eleq2 2898 . . . . . . . . . . . . . . . 16 (𝑡 = (𝑢 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑡 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣)))
26 sseq1 3989 . . . . . . . . . . . . . . . 16 (𝑡 = (𝑢 × 𝑣) → (𝑡𝑤 ↔ (𝑢 × 𝑣) ⊆ 𝑤))
2725, 26anbi12d 630 . . . . . . . . . . . . . . 15 (𝑡 = (𝑢 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑡𝑡𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
2824, 27rexrnmpo 7279 . . . . . . . . . . . . . 14 (∀𝑢𝐽𝑣𝐾 (𝑢 × 𝑣) ∈ V → (∃𝑡 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡𝑡𝑤) ↔ ∃𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
2923, 28ax-mp 5 . . . . . . . . . . . . 13 (∃𝑡 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡𝑡𝑤) ↔ ∃𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
3019, 29sylib 219 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ∃𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
3130ex 413 . . . . . . . . . . 11 (𝜑 → ((𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → ∃𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
32 r19.29 3251 . . . . . . . . . . . . 13 ((∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ ∃𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑢𝐽 (∀𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ ∃𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
33 r19.29 3251 . . . . . . . . . . . . . . 15 ((∀𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ ∃𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑣𝐾 (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
34 simprl 767 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣))
35 opelxp 5584 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑅𝑢𝑆𝑣))
3634, 35sylib 219 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (𝑅𝑢𝑆𝑣))
37 pm2.27 42 . . . . . . . . . . . . . . . . . . . . 21 (𝑅𝑢 → ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))
38 pm2.27 42 . . . . . . . . . . . . . . . . . . . . 21 (𝑆𝑣 → ((𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))
3937, 38im2anan9 619 . . . . . . . . . . . . . . . . . . . 20 ((𝑅𝑢𝑆𝑣) → (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
4036, 39syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
41 txlm.z . . . . . . . . . . . . . . . . . . . . 21 𝑍 = (ℤ𝑀)
4241rexanuz2 14697 . . . . . . . . . . . . . . . . . . . 20 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) ↔ (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))
4341uztrn2 12250 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
44 opelxpi 5585 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → ⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑢 × 𝑣))
45 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
46 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
4745, 46opeq12d 4803 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑘 → ⟨(𝐹𝑛), (𝐺𝑛)⟩ = ⟨(𝐹𝑘), (𝐺𝑘)⟩)
48 txlm.h . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝐻 = (𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)
49 opex 5347 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ V
5047, 48, 49fvmpt 6761 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘𝑍 → (𝐻𝑘) = ⟨(𝐹𝑘), (𝐺𝑘)⟩)
5150eleq1d 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘𝑍 → ((𝐻𝑘) ∈ (𝑢 × 𝑣) ↔ ⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑢 × 𝑣)))
5244, 51syl5ibr 247 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘𝑍 → (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → (𝐻𝑘) ∈ (𝑢 × 𝑣)))
5352adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘𝑍) → (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → (𝐻𝑘) ∈ (𝑢 × 𝑣)))
54 simplrr 774 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘𝑍) → (𝑢 × 𝑣) ⊆ 𝑤)
5554sseld 3963 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘𝑍) → ((𝐻𝑘) ∈ (𝑢 × 𝑣) → (𝐻𝑘) ∈ 𝑤))
5653, 55syld 47 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘𝑍) → (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → (𝐻𝑘) ∈ 𝑤))
5743, 56sylan2 592 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → (𝐻𝑘) ∈ 𝑤))
5857anassrs 468 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → (𝐻𝑘) ∈ 𝑤))
5958ralimdva 3174 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → ∀𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6059reximdva 3271 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6142, 60syl5bir 244 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ((∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6240, 61syld 47 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6362ex 413 . . . . . . . . . . . . . . . . 17 (((𝜑𝑢𝐽) ∧ 𝑣𝐾) → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
6463impcomd 412 . . . . . . . . . . . . . . . 16 (((𝜑𝑢𝐽) ∧ 𝑣𝐾) → ((((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6564rexlimdva 3281 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝐽) → (∃𝑣𝐾 (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6633, 65syl5 34 . . . . . . . . . . . . . 14 ((𝜑𝑢𝐽) → ((∀𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ ∃𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6766rexlimdva 3281 . . . . . . . . . . . . 13 (𝜑 → (∃𝑢𝐽 (∀𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ ∃𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6832, 67syl5 34 . . . . . . . . . . . 12 (𝜑 → ((∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ ∃𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6968expcomd 417 . . . . . . . . . . 11 (𝜑 → (∃𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
7031, 69syld 47 . . . . . . . . . 10 (𝜑 → ((𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → (∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
7170expdimp 453 . . . . . . . . 9 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 → (∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
7271com23 86 . . . . . . . 8 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → (∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
7372ralrimdva 3186 . . . . . . 7 (𝜑 → (∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
744, 73syl5 34 . . . . . 6 (𝜑 → ((∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
7574adantr 481 . . . . 5 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → ((∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
768adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → 𝐽 ∈ Top)
7711adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → 𝐾 ∈ Top)
78 simprr 769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → 𝑢𝐽)
79 toponmax 21462 . . . . . . . . . . . . . 14 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
809, 79syl 17 . . . . . . . . . . . . 13 (𝜑𝑌𝐾)
8180adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → 𝑌𝐾)
82 txopn 22138 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑢𝐽𝑌𝐾)) → (𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾))
8376, 77, 78, 81, 82syl22anc 834 . . . . . . . . . . 11 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → (𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾))
84 eleq2 2898 . . . . . . . . . . . . 13 (𝑤 = (𝑢 × 𝑌) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌)))
85 eleq2 2898 . . . . . . . . . . . . . 14 (𝑤 = (𝑢 × 𝑌) → ((𝐻𝑘) ∈ 𝑤 ↔ (𝐻𝑘) ∈ (𝑢 × 𝑌)))
8685rexralbidv 3298 . . . . . . . . . . . . 13 (𝑤 = (𝑢 × 𝑌) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌)))
8784, 86imbi12d 346 . . . . . . . . . . . 12 (𝑤 = (𝑢 × 𝑌) → ((⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌))))
8887rspcv 3615 . . . . . . . . . . 11 ((𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌))))
8983, 88syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌))))
90 simprl 767 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → 𝑆𝑌)
91 opelxpi 5585 . . . . . . . . . . . . 13 ((𝑅𝑢𝑆𝑌) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌))
9290, 91sylan2 592 . . . . . . . . . . . 12 ((𝑅𝑢 ∧ (𝜑 ∧ (𝑆𝑌𝑢𝐽))) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌))
9392expcom 414 . . . . . . . . . . 11 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → (𝑅𝑢 → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌)))
9450eleq1d 2894 . . . . . . . . . . . . . . . 16 (𝑘𝑍 → ((𝐻𝑘) ∈ (𝑢 × 𝑌) ↔ ⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑢 × 𝑌)))
95 opelxp1 5589 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑢 × 𝑌) → (𝐹𝑘) ∈ 𝑢)
9694, 95syl6bi 254 . . . . . . . . . . . . . . 15 (𝑘𝑍 → ((𝐻𝑘) ∈ (𝑢 × 𝑌) → (𝐹𝑘) ∈ 𝑢))
9743, 96syl 17 . . . . . . . . . . . . . 14 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → ((𝐻𝑘) ∈ (𝑢 × 𝑌) → (𝐹𝑘) ∈ 𝑢))
9897ralimdva 3174 . . . . . . . . . . . . 13 (𝑗𝑍 → (∀𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))
9998reximia 3239 . . . . . . . . . . . 12 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)
10099a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))
10193, 100imim12d 81 . . . . . . . . . 10 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌)) → (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
10289, 101syld 47 . . . . . . . . 9 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
103102anassrs 468 . . . . . . . 8 (((𝜑𝑆𝑌) ∧ 𝑢𝐽) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
104103ralrimdva 3186 . . . . . . 7 ((𝜑𝑆𝑌) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → ∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
105104adantrl 712 . . . . . 6 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → ∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
1068adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → 𝐽 ∈ Top)
10711adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → 𝐾 ∈ Top)
108 toponmax 21462 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1096, 108syl 17 . . . . . . . . . . . . 13 (𝜑𝑋𝐽)
110109adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → 𝑋𝐽)
111 simprr 769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → 𝑣𝐾)
112 txopn 22138 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑋𝐽𝑣𝐾)) → (𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾))
113106, 107, 110, 111, 112syl22anc 834 . . . . . . . . . . 11 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → (𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾))
114 eleq2 2898 . . . . . . . . . . . . 13 (𝑤 = (𝑋 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
115 eleq2 2898 . . . . . . . . . . . . . 14 (𝑤 = (𝑋 × 𝑣) → ((𝐻𝑘) ∈ 𝑤 ↔ (𝐻𝑘) ∈ (𝑋 × 𝑣)))
116115rexralbidv 3298 . . . . . . . . . . . . 13 (𝑤 = (𝑋 × 𝑣) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣)))
117114, 116imbi12d 346 . . . . . . . . . . . 12 (𝑤 = (𝑋 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣))))
118117rspcv 3615 . . . . . . . . . . 11 ((𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣))))
119113, 118syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣))))
120 opelxpi 5585 . . . . . . . . . . . . 13 ((𝑅𝑋𝑆𝑣) → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣))
121120ex 413 . . . . . . . . . . . 12 (𝑅𝑋 → (𝑆𝑣 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
122121ad2antrl 724 . . . . . . . . . . 11 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → (𝑆𝑣 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
12350eleq1d 2894 . . . . . . . . . . . . . . . 16 (𝑘𝑍 → ((𝐻𝑘) ∈ (𝑋 × 𝑣) ↔ ⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑋 × 𝑣)))
124 opelxp2 5590 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑋 × 𝑣) → (𝐺𝑘) ∈ 𝑣)
125123, 124syl6bi 254 . . . . . . . . . . . . . . 15 (𝑘𝑍 → ((𝐻𝑘) ∈ (𝑋 × 𝑣) → (𝐺𝑘) ∈ 𝑣))
12643, 125syl 17 . . . . . . . . . . . . . 14 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → ((𝐻𝑘) ∈ (𝑋 × 𝑣) → (𝐺𝑘) ∈ 𝑣))
127126ralimdva 3174 . . . . . . . . . . . . 13 (𝑗𝑍 → (∀𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣) → ∀𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))
128127reximia 3239 . . . . . . . . . . . 12 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)
129128a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))
130122, 129imim12d 81 . . . . . . . . . 10 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣)) → (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
131119, 130syld 47 . . . . . . . . 9 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
132131anassrs 468 . . . . . . . 8 (((𝜑𝑅𝑋) ∧ 𝑣𝐾) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
133132ralrimdva 3186 . . . . . . 7 ((𝜑𝑅𝑋) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
134133adantrr 713 . . . . . 6 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
135105, 134jcad 513 . . . . 5 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))))
13675, 135impbid 213 . . . 4 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → ((∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ↔ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
137136pm5.32da 579 . . 3 (𝜑 → (((𝑅𝑋𝑆𝑌) ∧ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))) ↔ ((𝑅𝑋𝑆𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))))
138 opelxp 5584 . . . 4 (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ↔ (𝑅𝑋𝑆𝑌))
139138anbi1i 623 . . 3 ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)) ↔ ((𝑅𝑋𝑆𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
140137, 139syl6bbr 290 . 2 (𝜑 → (((𝑅𝑋𝑆𝑌) ∧ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))))
141 txlm.m . . . . 5 (𝜑𝑀 ∈ ℤ)
142 txlm.f . . . . 5 (𝜑𝐹:𝑍𝑋)
143 eqidd 2819 . . . . 5 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
1446, 41, 141, 142, 143lmbrf 21796 . . . 4 (𝜑 → (𝐹(⇝𝑡𝐽)𝑅 ↔ (𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))))
145 txlm.g . . . . 5 (𝜑𝐺:𝑍𝑌)
146 eqidd 2819 . . . . 5 ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐺𝑘))
1479, 41, 141, 145, 146lmbrf 21796 . . . 4 (𝜑 → (𝐺(⇝𝑡𝐾)𝑆 ↔ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))))
148144, 147anbi12d 630 . . 3 (𝜑 → ((𝐹(⇝𝑡𝐽)𝑅𝐺(⇝𝑡𝐾)𝑆) ↔ ((𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) ∧ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))))
149 an4 652 . . 3 (((𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) ∧ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))) ↔ ((𝑅𝑋𝑆𝑌) ∧ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))))
150148, 149syl6bb 288 . 2 (𝜑 → ((𝐹(⇝𝑡𝐽)𝑅𝐺(⇝𝑡𝐾)𝑆) ↔ ((𝑅𝑋𝑆𝑌) ∧ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))))
151 txtopon 22127 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
1526, 9, 151syl2anc 584 . . 3 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
153142ffvelrnda 6843 . . . . 5 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ 𝑋)
154145ffvelrnda 6843 . . . . 5 ((𝜑𝑛𝑍) → (𝐺𝑛) ∈ 𝑌)
155153, 154opelxpd 5586 . . . 4 ((𝜑𝑛𝑍) → ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑋 × 𝑌))
156155, 48fmptd 6870 . . 3 (𝜑𝐻:𝑍⟶(𝑋 × 𝑌))
157 eqidd 2819 . . 3 ((𝜑𝑘𝑍) → (𝐻𝑘) = (𝐻𝑘))
158152, 41, 141, 156, 157lmbrf 21796 . 2 (𝜑 → (𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩ ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))))
159140, 150, 1583bitr4d 312 1 (𝜑 → ((𝐹(⇝𝑡𝐽)𝑅𝐺(⇝𝑡𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  wss 3933  cop 4563   class class class wbr 5057  cmpt 5137   × cxp 5546  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  cz 11969  cuz 12231  topGenctg 16699  Topctop 21429  TopOnctopon 21446  𝑡clm 21762   ×t ctx 22096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-er 8278  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-neg 10861  df-z 11970  df-uz 12232  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-lm 21765  df-tx 22098
This theorem is referenced by:  lmcn2  22185
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