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Theorem lmss 21150
 Description: Limit on a subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
Hypotheses
Ref Expression
lmss.1 𝐾 = (𝐽t 𝑌)
lmss.2 𝑍 = (ℤ𝑀)
lmss.3 (𝜑𝑌𝑉)
lmss.4 (𝜑𝐽 ∈ Top)
lmss.5 (𝜑𝑃𝑌)
lmss.6 (𝜑𝑀 ∈ ℤ)
lmss.7 (𝜑𝐹:𝑍𝑌)
Assertion
Ref Expression
lmss (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃))

Proof of Theorem lmss
Dummy variables 𝑗 𝑘 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmss.4 . . . . . 6 (𝜑𝐽 ∈ Top)
2 eqid 2651 . . . . . . 7 𝐽 = 𝐽
32toptopon 20770 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
41, 3sylib 208 . . . . 5 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
5 lmcl 21149 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝑃 𝐽)
64, 5sylan 487 . . . 4 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → 𝑃 𝐽)
7 lmfss 21148 . . . . . . 7 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝐽))
84, 7sylan 487 . . . . . 6 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝐽))
9 rnss 5386 . . . . . 6 (𝐹 ⊆ (ℂ × 𝐽) → ran 𝐹 ⊆ ran (ℂ × 𝐽))
108, 9syl 17 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → ran 𝐹 ⊆ ran (ℂ × 𝐽))
11 rnxpss 5601 . . . . 5 ran (ℂ × 𝐽) ⊆ 𝐽
1210, 11syl6ss 3648 . . . 4 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → ran 𝐹 𝐽)
136, 12jca 553 . . 3 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → (𝑃 𝐽 ∧ ran 𝐹 𝐽))
1413ex 449 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 → (𝑃 𝐽 ∧ ran 𝐹 𝐽)))
15 inss2 3867 . . . . 5 (𝑌 𝐽) ⊆ 𝐽
16 lmss.1 . . . . . . 7 𝐾 = (𝐽t 𝑌)
17 lmss.3 . . . . . . . 8 (𝜑𝑌𝑉)
18 resttopon2 21020 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑌𝑉) → (𝐽t 𝑌) ∈ (TopOn‘(𝑌 𝐽)))
194, 17, 18syl2anc 694 . . . . . . 7 (𝜑 → (𝐽t 𝑌) ∈ (TopOn‘(𝑌 𝐽)))
2016, 19syl5eqel 2734 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘(𝑌 𝐽)))
21 lmcl 21149 . . . . . 6 ((𝐾 ∈ (TopOn‘(𝑌 𝐽)) ∧ 𝐹(⇝𝑡𝐾)𝑃) → 𝑃 ∈ (𝑌 𝐽))
2220, 21sylan 487 . . . . 5 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → 𝑃 ∈ (𝑌 𝐽))
2315, 22sseldi 3634 . . . 4 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → 𝑃 𝐽)
24 lmfss 21148 . . . . . . . 8 ((𝐾 ∈ (TopOn‘(𝑌 𝐽)) ∧ 𝐹(⇝𝑡𝐾)𝑃) → 𝐹 ⊆ (ℂ × (𝑌 𝐽)))
2520, 24sylan 487 . . . . . . 7 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → 𝐹 ⊆ (ℂ × (𝑌 𝐽)))
26 rnss 5386 . . . . . . 7 (𝐹 ⊆ (ℂ × (𝑌 𝐽)) → ran 𝐹 ⊆ ran (ℂ × (𝑌 𝐽)))
2725, 26syl 17 . . . . . 6 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → ran 𝐹 ⊆ ran (ℂ × (𝑌 𝐽)))
28 rnxpss 5601 . . . . . 6 ran (ℂ × (𝑌 𝐽)) ⊆ (𝑌 𝐽)
2927, 28syl6ss 3648 . . . . 5 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → ran 𝐹 ⊆ (𝑌 𝐽))
3029, 15syl6ss 3648 . . . 4 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → ran 𝐹 𝐽)
3123, 30jca 553 . . 3 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → (𝑃 𝐽 ∧ ran 𝐹 𝐽))
3231ex 449 . 2 (𝜑 → (𝐹(⇝𝑡𝐾)𝑃 → (𝑃 𝐽 ∧ ran 𝐹 𝐽)))
33 simprl 809 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑃 𝐽)
34 lmss.5 . . . . . . . 8 (𝜑𝑃𝑌)
3534adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑃𝑌)
3635, 33elind 3831 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑃 ∈ (𝑌 𝐽))
3733, 362thd 255 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑃 𝐽𝑃 ∈ (𝑌 𝐽)))
3816eleq2i 2722 . . . . . . . . 9 (𝑣𝐾𝑣 ∈ (𝐽t 𝑌))
391adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐽 ∈ Top)
4017adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑌𝑉)
41 elrest 16135 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑌𝑉) → (𝑣 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑣 = (𝑢𝑌)))
4239, 40, 41syl2anc 694 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑣 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑣 = (𝑢𝑌)))
4342biimpa 500 . . . . . . . . 9 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑣 ∈ (𝐽t 𝑌)) → ∃𝑢𝐽 𝑣 = (𝑢𝑌))
4438, 43sylan2b 491 . . . . . . . 8 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑣𝐾) → ∃𝑢𝐽 𝑣 = (𝑢𝑌))
45 r19.29r 3102 . . . . . . . . . 10 ((∃𝑢𝐽 𝑣 = (𝑢𝑌) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → ∃𝑢𝐽 (𝑣 = (𝑢𝑌) ∧ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
4635biantrud 527 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑃𝑢 ↔ (𝑃𝑢𝑃𝑌)))
47 elin 3829 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ (𝑢𝑌) ↔ (𝑃𝑢𝑃𝑌))
4846, 47syl6bbr 278 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑃𝑢𝑃 ∈ (𝑢𝑌)))
49 lmss.2 . . . . . . . . . . . . . . . . . . . . 21 𝑍 = (ℤ𝑀)
5049uztrn2 11743 . . . . . . . . . . . . . . . . . . . 20 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
51 lmss.7 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹:𝑍𝑌)
5251adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹:𝑍𝑌)
5352ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → (𝐹𝑘) ∈ 𝑌)
5453biantrud 527 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → ((𝐹𝑘) ∈ 𝑢 ↔ ((𝐹𝑘) ∈ 𝑢 ∧ (𝐹𝑘) ∈ 𝑌)))
55 elin 3829 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑘) ∈ (𝑢𝑌) ↔ ((𝐹𝑘) ∈ 𝑢 ∧ (𝐹𝑘) ∈ 𝑌))
5654, 55syl6bbr 278 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → ((𝐹𝑘) ∈ 𝑢 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
5750, 56sylan2 490 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) ∈ 𝑢 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
5857anassrs 681 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝐹𝑘) ∈ 𝑢 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
5958ralbidva 3014 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))
6059rexbidva 3078 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))
6148, 60imbi12d 333 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ↔ (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
6261adantr 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ↔ (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
6362biimpd 219 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
64 eleq2 2719 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢𝑌) → (𝑃𝑣𝑃 ∈ (𝑢𝑌)))
65 eleq2 2719 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑢𝑌) → ((𝐹𝑘) ∈ 𝑣 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
6665rexralbidv 3087 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢𝑌) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))
6764, 66imbi12d 333 . . . . . . . . . . . . . 14 (𝑣 = (𝑢𝑌) → ((𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) ↔ (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
6867imbi2d 329 . . . . . . . . . . . . 13 (𝑣 = (𝑢𝑌) → (((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)) ↔ ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))))
6963, 68syl5ibrcom 237 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (𝑣 = (𝑢𝑌) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣))))
7069impd 446 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → ((𝑣 = (𝑢𝑌) ∧ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7170rexlimdva 3060 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∃𝑢𝐽 (𝑣 = (𝑢𝑌) ∧ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7245, 71syl5 34 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ((∃𝑢𝐽 𝑣 = (𝑢𝑌) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7372expdimp 452 . . . . . . . 8 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ ∃𝑢𝐽 𝑣 = (𝑢𝑌)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7444, 73syldan 486 . . . . . . 7 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑣𝐾) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7574ralrimdva 2998 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7639adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → 𝐽 ∈ Top)
7740adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → 𝑌𝑉)
78 simpr 476 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → 𝑢𝐽)
79 elrestr 16136 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑌𝑉𝑢𝐽) → (𝑢𝑌) ∈ (𝐽t 𝑌))
8076, 77, 78, 79syl3anc 1366 . . . . . . . . . 10 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ (𝐽t 𝑌))
8180, 16syl6eleqr 2741 . . . . . . . . 9 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ 𝐾)
8267rspcv 3336 . . . . . . . . 9 ((𝑢𝑌) ∈ 𝐾 → (∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) → (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
8381, 82syl 17 . . . . . . . 8 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) → (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
8483, 62sylibrd 249 . . . . . . 7 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) → (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
8584ralrimdva 2998 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
8675, 85impbid 202 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ↔ ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
8737, 86anbi12d 747 . . . 4 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ((𝑃 𝐽 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) ↔ (𝑃 ∈ (𝑌 𝐽) ∧ ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣))))
8839, 3sylib 208 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐽))
89 lmss.6 . . . . . 6 (𝜑𝑀 ∈ ℤ)
9089adantr 480 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑀 ∈ ℤ)
91 ffn 6083 . . . . . . 7 (𝐹:𝑍𝑌𝐹 Fn 𝑍)
9252, 91syl 17 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹 Fn 𝑍)
93 simprr 811 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ran 𝐹 𝐽)
94 df-f 5930 . . . . . 6 (𝐹:𝑍 𝐽 ↔ (𝐹 Fn 𝑍 ∧ ran 𝐹 𝐽))
9592, 93, 94sylanbrc 699 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹:𝑍 𝐽)
96 eqidd 2652 . . . . 5 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
9788, 49, 90, 95, 96lmbrf 21112 . . . 4 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝑃 𝐽 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))))
9820adantr 480 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐾 ∈ (TopOn‘(𝑌 𝐽)))
99 frn 6091 . . . . . . . 8 (𝐹:𝑍𝑌 → ran 𝐹𝑌)
10052, 99syl 17 . . . . . . 7 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ran 𝐹𝑌)
101100, 93ssind 3870 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ran 𝐹 ⊆ (𝑌 𝐽))
102 df-f 5930 . . . . . 6 (𝐹:𝑍⟶(𝑌 𝐽) ↔ (𝐹 Fn 𝑍 ∧ ran 𝐹 ⊆ (𝑌 𝐽)))
10392, 101, 102sylanbrc 699 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹:𝑍⟶(𝑌 𝐽))
10498, 49, 90, 103, 96lmbrf 21112 . . . 4 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝐹(⇝𝑡𝐾)𝑃 ↔ (𝑃 ∈ (𝑌 𝐽) ∧ ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣))))
10587, 97, 1043bitr4d 300 . . 3 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃))
106105ex 449 . 2 (𝜑 → ((𝑃 𝐽 ∧ ran 𝐹 𝐽) → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃)))
10714, 32, 106pm5.21ndd 368 1 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ∩ cin 3606   ⊆ wss 3607  ∪ cuni 4468   class class class wbr 4685   × cxp 5141  ran crn 5144   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  ℂcc 9972  ℤcz 11415  ℤ≥cuz 11725   ↾t crest 16128  Topctop 20746  TopOnctopon 20763  ⇝𝑡clm 21078 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-er 7787  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-neg 10307  df-z 11416  df-uz 11726  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-lm 21081 This theorem is referenced by:  1stckgen  21405  minvecolem4b  27862  minvecolem4  27864  hhsscms  28264  lmlim  30121  climreeq  40163  xlimclim  40368
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