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Theorem txlm 21672
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 3218 . . . . . . . 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 3219 . . . . . . . . 9 (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∀𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
32ralimi 3101 . . . . . . . 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 754 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → 𝑤 ∈ (𝐽 ×t 𝐾))
6 txlm.j . . . . . . . . . . . . . . . . . 18 (𝜑𝐽 ∈ (TopOn‘𝑋))
7 topontop 20938 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
86, 7syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐽 ∈ Top)
9 txlm.k . . . . . . . . . . . . . . . . . 18 (𝜑𝐾 ∈ (TopOn‘𝑌))
10 topontop 20938 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
119, 10syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ Top)
12 eqid 2771 . . . . . . . . . . . . . . . . . 18 ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)) = ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))
1312txval 21588 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
148, 11, 13syl2anc 573 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
1514adantr 466 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
165, 15eleqtrd 2852 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → 𝑤 ∈ (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))))
17 simprr 756 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ⟨𝑅, 𝑆⟩ ∈ 𝑤)
18 tg2 20990 . . . . . . . . . . . . . 14 ((𝑤 ∈ (topGen‘ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → ∃𝑡 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡𝑡𝑤))
1916, 17, 18syl2anc 573 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ∃𝑡 ∈ ran (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))(⟨𝑅, 𝑆⟩ ∈ 𝑡𝑡𝑤))
20 vex 3354 . . . . . . . . . . . . . . . 16 𝑢 ∈ V
21 vex 3354 . . . . . . . . . . . . . . . 16 𝑣 ∈ V
2220, 21xpex 7109 . . . . . . . . . . . . . . 15 (𝑢 × 𝑣) ∈ V
2322rgen2w 3074 . . . . . . . . . . . . . 14 𝑢𝐽𝑣𝐾 (𝑢 × 𝑣) ∈ V
24 eqid 2771 . . . . . . . . . . . . . . 15 (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣)) = (𝑢𝐽, 𝑣𝐾 ↦ (𝑢 × 𝑣))
25 eleq2 2839 . . . . . . . . . . . . . . . 16 (𝑡 = (𝑢 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑡 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣)))
26 sseq1 3775 . . . . . . . . . . . . . . . 16 (𝑡 = (𝑢 × 𝑣) → (𝑡𝑤 ↔ (𝑢 × 𝑣) ⊆ 𝑤))
2725, 26anbi12d 616 . . . . . . . . . . . . . . 15 (𝑡 = (𝑢 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑡𝑡𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
2824, 27rexrnmpt2 6923 . . . . . . . . . . . . . 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 208 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤)) → ∃𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
3130ex 397 . . . . . . . . . . 11 (𝜑 → ((𝑤 ∈ (𝐽 ×t 𝐾) ∧ ⟨𝑅, 𝑆⟩ ∈ 𝑤) → ∃𝑢𝐽𝑣𝐾 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
32 r19.29 3220 . . . . . . . . . . . . 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 3220 . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣))
35 opelxp 5286 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ↔ (𝑅𝑢𝑆𝑣))
3634, 35sylib 208 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (𝑅𝑢𝑆𝑣))
37 pm2.27 42 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅𝑢 → ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))
38 pm2.27 42 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆𝑣 → ((𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))
3937, 38im2anan9 606 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅𝑢𝑆𝑣) → (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
4036, 39syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((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 . . . . . . . . . . . . . . . . . . . . . 22 𝑍 = (ℤ𝑀)
4241rexanuz2 14297 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) ↔ (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))
4341uztrn2 11906 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
44 opelxpi 5288 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → ⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑢 × 𝑣))
45 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
46 fveq2 6332 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
4745, 46opeq12d 4547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝑘 → ⟨(𝐹𝑛), (𝐺𝑛)⟩ = ⟨(𝐹𝑘), (𝐺𝑘)⟩)
48 txlm.h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝐻 = (𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)
49 opex 5060 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ V
5047, 48, 49fvmpt 6424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘𝑍 → (𝐻𝑘) = ⟨(𝐹𝑘), (𝐺𝑘)⟩)
5150eleq1d 2835 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘𝑍 → ((𝐻𝑘) ∈ (𝑢 × 𝑣) ↔ ⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑢 × 𝑣)))
5244, 51syl5ibr 236 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘𝑍 → (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → (𝐻𝑘) ∈ (𝑢 × 𝑣)))
5352adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘𝑍) → (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → (𝐻𝑘) ∈ (𝑢 × 𝑣)))
54 simplrr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘𝑍) → (𝑢 × 𝑣) ⊆ 𝑤)
5554sseld 3751 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘𝑍) → ((𝐻𝑘) ∈ (𝑢 × 𝑣) → (𝐻𝑘) ∈ 𝑤))
5653, 55syld 47 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑘𝑍) → (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → (𝐻𝑘) ∈ 𝑤))
5743, 56sylan2 580 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → (𝐻𝑘) ∈ 𝑤))
5857anassrs 458 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → (𝐻𝑘) ∈ 𝑤))
5958ralimdva 3111 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) ∧ 𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → ∀𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6059reximdva 3165 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ 𝑢 ∧ (𝐺𝑘) ∈ 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6142, 60syl5bir 233 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ((∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ∧ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6240, 61syld 47 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑢𝐽) ∧ 𝑣𝐾) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6362ex 397 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑢𝐽) ∧ 𝑣𝐾) → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
6463com23 86 . . . . . . . . . . . . . . . . 17 (((𝜑𝑢𝐽) ∧ 𝑣𝐾) → (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
6564impd 396 . . . . . . . . . . . . . . . 16 (((𝜑𝑢𝐽) ∧ 𝑣𝐾) → ((((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6665rexlimdva 3179 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝐽) → (∃𝑣𝐾 (((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) ∧ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))
6733, 66syl5 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))
6867rexlimdva 3179 . . . . . . . . . . . . 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))
6932, 68syl5 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))
7069expcomd 402 . . . . . . . . . . 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)))
7131, 70syld 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)))
7271expdimp 440 . . . . . . . . 9 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 → (∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
7372com23 86 . . . . . . . 8 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → (∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
7473ralrimdva 3118 . . . . . . 7 (𝜑 → (∀𝑢𝐽𝑣𝐾 ((𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)) → ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤)))
754, 74syl5 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)))
7675adantr 466 . . . . 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)))
778adantr 466 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → 𝐽 ∈ Top)
7811adantr 466 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → 𝐾 ∈ Top)
79 simprr 756 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → 𝑢𝐽)
80 toponmax 20951 . . . . . . . . . . . . . 14 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
819, 80syl 17 . . . . . . . . . . . . 13 (𝜑𝑌𝐾)
8281adantr 466 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → 𝑌𝐾)
83 txopn 21626 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑢𝐽𝑌𝐾)) → (𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾))
8477, 78, 79, 82, 83syl22anc 1477 . . . . . . . . . . 11 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → (𝑢 × 𝑌) ∈ (𝐽 ×t 𝐾))
85 eleq2 2839 . . . . . . . . . . . . 13 (𝑤 = (𝑢 × 𝑌) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌)))
86 eleq2 2839 . . . . . . . . . . . . . 14 (𝑤 = (𝑢 × 𝑌) → ((𝐻𝑘) ∈ 𝑤 ↔ (𝐻𝑘) ∈ (𝑢 × 𝑌)))
8786rexralbidv 3206 . . . . . . . . . . . . 13 (𝑤 = (𝑢 × 𝑌) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌)))
8885, 87imbi12d 333 . . . . . . . . . . . 12 (𝑤 = (𝑢 × 𝑌) → ((⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌))))
8988rspcv 3456 . . . . . . . . . . 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))))
9084, 89syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌))))
91 simprl 754 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → 𝑆𝑌)
92 opelxpi 5288 . . . . . . . . . . . . 13 ((𝑅𝑢𝑆𝑌) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌))
9391, 92sylan2 580 . . . . . . . . . . . 12 ((𝑅𝑢 ∧ (𝜑 ∧ (𝑆𝑌𝑢𝐽))) → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌))
9493expcom 398 . . . . . . . . . . 11 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → (𝑅𝑢 → ⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌)))
9550eleq1d 2835 . . . . . . . . . . . . . . . 16 (𝑘𝑍 → ((𝐻𝑘) ∈ (𝑢 × 𝑌) ↔ ⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑢 × 𝑌)))
96 opelxp1 5290 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑢 × 𝑌) → (𝐹𝑘) ∈ 𝑢)
9795, 96syl6bi 243 . . . . . . . . . . . . . . 15 (𝑘𝑍 → ((𝐻𝑘) ∈ (𝑢 × 𝑌) → (𝐹𝑘) ∈ 𝑢))
9843, 97syl 17 . . . . . . . . . . . . . 14 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → ((𝐻𝑘) ∈ (𝑢 × 𝑌) → (𝐹𝑘) ∈ 𝑢))
9998ralimdva 3111 . . . . . . . . . . . . 13 (𝑗𝑍 → (∀𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌) → ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))
10099reximia 3157 . . . . . . . . . . . 12 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)
101100a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))
10294, 101imim12d 81 . . . . . . . . . 10 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → ((⟨𝑅, 𝑆⟩ ∈ (𝑢 × 𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑢 × 𝑌)) → (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
10390, 102syld 47 . . . . . . . . 9 ((𝜑 ∧ (𝑆𝑌𝑢𝐽)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
104103anassrs 458 . . . . . . . 8 (((𝜑𝑆𝑌) ∧ 𝑢𝐽) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
105104ralrimdva 3118 . . . . . . 7 ((𝜑𝑆𝑌) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → ∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
106105adantrl 695 . . . . . 6 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → ∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
1078adantr 466 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → 𝐽 ∈ Top)
10811adantr 466 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → 𝐾 ∈ Top)
109 toponmax 20951 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
1106, 109syl 17 . . . . . . . . . . . . 13 (𝜑𝑋𝐽)
111110adantr 466 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → 𝑋𝐽)
112 simprr 756 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → 𝑣𝐾)
113 txopn 21626 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑋𝐽𝑣𝐾)) → (𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾))
114107, 108, 111, 112, 113syl22anc 1477 . . . . . . . . . . 11 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → (𝑋 × 𝑣) ∈ (𝐽 ×t 𝐾))
115 eleq2 2839 . . . . . . . . . . . . 13 (𝑤 = (𝑋 × 𝑣) → (⟨𝑅, 𝑆⟩ ∈ 𝑤 ↔ ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
116 eleq2 2839 . . . . . . . . . . . . . 14 (𝑤 = (𝑋 × 𝑣) → ((𝐻𝑘) ∈ 𝑤 ↔ (𝐻𝑘) ∈ (𝑋 × 𝑣)))
117116rexralbidv 3206 . . . . . . . . . . . . 13 (𝑤 = (𝑋 × 𝑣) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣)))
118115, 117imbi12d 333 . . . . . . . . . . . 12 (𝑤 = (𝑋 × 𝑣) → ((⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣))))
119118rspcv 3456 . . . . . . . . . . 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))))
120114, 119syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣))))
121 opelxpi 5288 . . . . . . . . . . . . 13 ((𝑅𝑋𝑆𝑣) → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣))
122121ex 397 . . . . . . . . . . . 12 (𝑅𝑋 → (𝑆𝑣 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
123122ad2antrl 707 . . . . . . . . . . 11 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → (𝑆𝑣 → ⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣)))
12450eleq1d 2835 . . . . . . . . . . . . . . . 16 (𝑘𝑍 → ((𝐻𝑘) ∈ (𝑋 × 𝑣) ↔ ⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑋 × 𝑣)))
125 opelxp2 5291 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑘), (𝐺𝑘)⟩ ∈ (𝑋 × 𝑣) → (𝐺𝑘) ∈ 𝑣)
126124, 125syl6bi 243 . . . . . . . . . . . . . . 15 (𝑘𝑍 → ((𝐻𝑘) ∈ (𝑋 × 𝑣) → (𝐺𝑘) ∈ 𝑣))
12743, 126syl 17 . . . . . . . . . . . . . 14 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → ((𝐻𝑘) ∈ (𝑋 × 𝑣) → (𝐺𝑘) ∈ 𝑣))
128127ralimdva 3111 . . . . . . . . . . . . 13 (𝑗𝑍 → (∀𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣) → ∀𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))
129128reximia 3157 . . . . . . . . . . . 12 (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)
130129a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))
131123, 130imim12d 81 . . . . . . . . . 10 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → ((⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑣) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ (𝑋 × 𝑣)) → (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
132120, 131syld 47 . . . . . . . . 9 ((𝜑 ∧ (𝑅𝑋𝑣𝐾)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
133132anassrs 458 . . . . . . . 8 (((𝜑𝑅𝑋) ∧ 𝑣𝐾) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
134133ralrimdva 3118 . . . . . . 7 ((𝜑𝑅𝑋) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
135134adantrr 696 . . . . . 6 ((𝜑 ∧ (𝑅𝑋𝑆𝑌)) → (∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤) → ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))
136106, 135jcad 502 . . . . 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))))
13776, 136impbid 202 . . . 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)))
138137pm5.32da 568 . . 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))))
139 opelxp 5286 . . . 4 (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ↔ (𝑅𝑋𝑆𝑌))
140139anbi1i 610 . . 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)))
141138, 140syl6bbr 278 . 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))))
142 txlm.m . . . . 5 (𝜑𝑀 ∈ ℤ)
143 txlm.f . . . . 5 (𝜑𝐹:𝑍𝑋)
144 eqidd 2772 . . . . 5 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
1456, 41, 142, 143, 144lmbrf 21285 . . . 4 (𝜑 → (𝐹(⇝𝑡𝐽)𝑅 ↔ (𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))))
146 txlm.g . . . . 5 (𝜑𝐺:𝑍𝑌)
147 eqidd 2772 . . . . 5 ((𝜑𝑘𝑍) → (𝐺𝑘) = (𝐺𝑘))
1489, 41, 142, 146, 147lmbrf 21285 . . . 4 (𝜑 → (𝐺(⇝𝑡𝐾)𝑆 ↔ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣))))
149145, 148anbi12d 616 . . 3 (𝜑 → ((𝐹(⇝𝑡𝐽)𝑅𝐺(⇝𝑡𝐾)𝑆) ↔ ((𝑅𝑋 ∧ ∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) ∧ (𝑆𝑌 ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))))
150 an4 635 . . 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))))
151149, 150syl6bb 276 . 2 (𝜑 → ((𝐹(⇝𝑡𝐽)𝑅𝐺(⇝𝑡𝐾)𝑆) ↔ ((𝑅𝑋𝑆𝑌) ∧ (∀𝑢𝐽 (𝑅𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ∧ ∀𝑣𝐾 (𝑆𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐺𝑘) ∈ 𝑣)))))
152 txtopon 21615 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
1536, 9, 152syl2anc 573 . . 3 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
154143ffvelrnda 6502 . . . . 5 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ 𝑋)
155146ffvelrnda 6502 . . . . 5 ((𝜑𝑛𝑍) → (𝐺𝑛) ∈ 𝑌)
156 opelxpi 5288 . . . . 5 (((𝐹𝑛) ∈ 𝑋 ∧ (𝐺𝑛) ∈ 𝑌) → ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑋 × 𝑌))
157154, 155, 156syl2anc 573 . . . 4 ((𝜑𝑛𝑍) → ⟨(𝐹𝑛), (𝐺𝑛)⟩ ∈ (𝑋 × 𝑌))
158157, 48fmptd 6527 . . 3 (𝜑𝐻:𝑍⟶(𝑋 × 𝑌))
159 eqidd 2772 . . 3 ((𝜑𝑘𝑍) → (𝐻𝑘) = (𝐻𝑘))
160153, 41, 142, 158, 159lmbrf 21285 . 2 (𝜑 → (𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩ ↔ (⟨𝑅, 𝑆⟩ ∈ (𝑋 × 𝑌) ∧ ∀𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝑅, 𝑆⟩ ∈ 𝑤 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐻𝑘) ∈ 𝑤))))
161141, 151, 1603bitr4d 300 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 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  wss 3723  cop 4322   class class class wbr 4786  cmpt 4863   × cxp 5247  ran crn 5250  wf 6027  cfv 6031  (class class class)co 6793  cmpt2 6795  cz 11579  cuz 11888  topGenctg 16306  Topctop 20918  TopOnctopon 20935  𝑡clm 21251   ×t ctx 21584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-pre-lttri 10212  ax-pre-lttrn 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-er 7896  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-neg 10471  df-z 11580  df-uz 11889  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-lm 21254  df-tx 21586
This theorem is referenced by:  lmcn2  21673
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