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Theorem lmss 20851
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 2606 . . . . . . 7 𝐽 = 𝐽
32toptopon 20487 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
41, 3sylib 206 . . . . 5 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
5 lmcl 20850 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝑃 𝐽)
64, 5sylan 486 . . . 4 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → 𝑃 𝐽)
7 lmfss 20849 . . . . . . 7 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝐽))
84, 7sylan 486 . . . . . 6 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → 𝐹 ⊆ (ℂ × 𝐽))
9 rnss 5259 . . . . . 6 (𝐹 ⊆ (ℂ × 𝐽) → ran 𝐹 ⊆ ran (ℂ × 𝐽))
108, 9syl 17 . . . . 5 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → ran 𝐹 ⊆ ran (ℂ × 𝐽))
11 rnxpss 5468 . . . . 5 ran (ℂ × 𝐽) ⊆ 𝐽
1210, 11syl6ss 3576 . . . 4 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → ran 𝐹 𝐽)
136, 12jca 552 . . 3 ((𝜑𝐹(⇝𝑡𝐽)𝑃) → (𝑃 𝐽 ∧ ran 𝐹 𝐽))
1413ex 448 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 → (𝑃 𝐽 ∧ ran 𝐹 𝐽)))
15 inss2 3792 . . . . 5 (𝑌 𝐽) ⊆ 𝐽
16 lmss.1 . . . . . . 7 𝐾 = (𝐽t 𝑌)
17 lmss.3 . . . . . . . 8 (𝜑𝑌𝑉)
18 resttopon2 20721 . . . . . . . 8 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑌𝑉) → (𝐽t 𝑌) ∈ (TopOn‘(𝑌 𝐽)))
194, 17, 18syl2anc 690 . . . . . . 7 (𝜑 → (𝐽t 𝑌) ∈ (TopOn‘(𝑌 𝐽)))
2016, 19syl5eqel 2688 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘(𝑌 𝐽)))
21 lmcl 20850 . . . . . 6 ((𝐾 ∈ (TopOn‘(𝑌 𝐽)) ∧ 𝐹(⇝𝑡𝐾)𝑃) → 𝑃 ∈ (𝑌 𝐽))
2220, 21sylan 486 . . . . 5 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → 𝑃 ∈ (𝑌 𝐽))
2315, 22sseldi 3562 . . . 4 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → 𝑃 𝐽)
24 lmfss 20849 . . . . . . . 8 ((𝐾 ∈ (TopOn‘(𝑌 𝐽)) ∧ 𝐹(⇝𝑡𝐾)𝑃) → 𝐹 ⊆ (ℂ × (𝑌 𝐽)))
2520, 24sylan 486 . . . . . . 7 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → 𝐹 ⊆ (ℂ × (𝑌 𝐽)))
26 rnss 5259 . . . . . . 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 5468 . . . . . 6 ran (ℂ × (𝑌 𝐽)) ⊆ (𝑌 𝐽)
2927, 28syl6ss 3576 . . . . 5 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → ran 𝐹 ⊆ (𝑌 𝐽))
3029, 15syl6ss 3576 . . . 4 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → ran 𝐹 𝐽)
3123, 30jca 552 . . 3 ((𝜑𝐹(⇝𝑡𝐾)𝑃) → (𝑃 𝐽 ∧ ran 𝐹 𝐽))
3231ex 448 . 2 (𝜑 → (𝐹(⇝𝑡𝐾)𝑃 → (𝑃 𝐽 ∧ ran 𝐹 𝐽)))
33 simprl 789 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑃 𝐽)
34 lmss.5 . . . . . . . 8 (𝜑𝑃𝑌)
3534adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑃𝑌)
3635, 33elind 3756 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑃 ∈ (𝑌 𝐽))
3733, 362thd 253 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑃 𝐽𝑃 ∈ (𝑌 𝐽)))
3816eleq2i 2676 . . . . . . . . 9 (𝑣𝐾𝑣 ∈ (𝐽t 𝑌))
391adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐽 ∈ Top)
4017adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑌𝑉)
41 elrest 15854 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑌𝑉) → (𝑣 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑣 = (𝑢𝑌)))
4239, 40, 41syl2anc 690 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑣 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑣 = (𝑢𝑌)))
4342biimpa 499 . . . . . . . . 9 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑣 ∈ (𝐽t 𝑌)) → ∃𝑢𝐽 𝑣 = (𝑢𝑌))
4438, 43sylan2b 490 . . . . . . . 8 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑣𝐾) → ∃𝑢𝐽 𝑣 = (𝑢𝑌))
45 r19.29r 3051 . . . . . . . . . 10 ((∃𝑢𝐽 𝑣 = (𝑢𝑌) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → ∃𝑢𝐽 (𝑣 = (𝑢𝑌) ∧ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
4635biantrud 526 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑃𝑢 ↔ (𝑃𝑢𝑃𝑌)))
47 elin 3754 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ (𝑢𝑌) ↔ (𝑃𝑢𝑃𝑌))
4846, 47syl6bbr 276 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝑃𝑢𝑃 ∈ (𝑢𝑌)))
49 lmss.2 . . . . . . . . . . . . . . . . . . . . 21 𝑍 = (ℤ𝑀)
5049uztrn2 11534 . . . . . . . . . . . . . . . . . . . 20 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
51 lmss.7 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹:𝑍𝑌)
5251adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹:𝑍𝑌)
5352ffvelrnda 6249 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → (𝐹𝑘) ∈ 𝑌)
5453biantrud 526 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → ((𝐹𝑘) ∈ 𝑢 ↔ ((𝐹𝑘) ∈ 𝑢 ∧ (𝐹𝑘) ∈ 𝑌)))
55 elin 3754 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑘) ∈ (𝑢𝑌) ↔ ((𝐹𝑘) ∈ 𝑢 ∧ (𝐹𝑘) ∈ 𝑌))
5654, 55syl6bbr 276 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → ((𝐹𝑘) ∈ 𝑢 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
5750, 56sylan2 489 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) ∈ 𝑢 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
5857anassrs 677 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((𝐹𝑘) ∈ 𝑢 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
5958ralbidva 2964 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑗𝑍) → (∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ↔ ∀𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))
6059rexbidva 3027 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))
6148, 60imbi12d 332 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ↔ (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
6261adantr 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ↔ (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
6362biimpd 217 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
64 eleq2 2673 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢𝑌) → (𝑃𝑣𝑃 ∈ (𝑢𝑌)))
65 eleq2 2673 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑢𝑌) → ((𝐹𝑘) ∈ 𝑣 ↔ (𝐹𝑘) ∈ (𝑢𝑌)))
6665rexralbidv 3036 . . . . . . . . . . . . . . 15 (𝑣 = (𝑢𝑌) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌)))
6764, 66imbi12d 332 . . . . . . . . . . . . . 14 (𝑣 = (𝑢𝑌) → ((𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) ↔ (𝑃 ∈ (𝑢𝑌) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ (𝑢𝑌))))
6867imbi2d 328 . . . . . . . . . . . . 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 235 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (𝑣 = (𝑢𝑌) → ((𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣))))
7069impd 445 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → ((𝑣 = (𝑢𝑌) ∧ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7170rexlimdva 3009 . . . . . . . . . 10 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∃𝑢𝐽 (𝑣 = (𝑢𝑌) ∧ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7245, 71syl5 33 . . . . . . . . 9 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ((∃𝑢𝐽 𝑣 = (𝑢𝑌) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7372expdimp 451 . . . . . . . 8 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ ∃𝑢𝐽 𝑣 = (𝑢𝑌)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7444, 73syldan 485 . . . . . . 7 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑣𝐾) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7574ralrimdva 2948 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) → ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
7639adantr 479 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → 𝐽 ∈ Top)
7740adantr 479 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → 𝑌𝑉)
78 simpr 475 . . . . . . . . . . 11 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → 𝑢𝐽)
79 elrestr 15855 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑌𝑉𝑢𝐽) → (𝑢𝑌) ∈ (𝐽t 𝑌))
8076, 77, 78, 79syl3anc 1317 . . . . . . . . . 10 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ (𝐽t 𝑌))
8180, 16syl6eleqr 2695 . . . . . . . . 9 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (𝑢𝑌) ∈ 𝐾)
8267rspcv 3274 . . . . . . . . 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 247 . . . . . . 7 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑢𝐽) → (∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) → (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
8584ralrimdva 2948 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣) → ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢)))
8675, 85impbid 200 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢) ↔ ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣)))
8737, 86anbi12d 742 . . . 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 206 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐽))
89 lmss.6 . . . . . 6 (𝜑𝑀 ∈ ℤ)
9089adantr 479 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝑀 ∈ ℤ)
91 ffn 5941 . . . . . . 7 (𝐹:𝑍𝑌𝐹 Fn 𝑍)
9252, 91syl 17 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹 Fn 𝑍)
93 simprr 791 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ran 𝐹 𝐽)
94 df-f 5791 . . . . . 6 (𝐹:𝑍 𝐽 ↔ (𝐹 Fn 𝑍 ∧ ran 𝐹 𝐽))
9592, 93, 94sylanbrc 694 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹:𝑍 𝐽)
96 eqidd 2607 . . . . 5 (((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) ∧ 𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
9788, 49, 90, 95, 96lmbrf 20813 . . . 4 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝑃 𝐽 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑢))))
9820adantr 479 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐾 ∈ (TopOn‘(𝑌 𝐽)))
99 frn 5949 . . . . . . . 8 (𝐹:𝑍𝑌 → ran 𝐹𝑌)
10052, 99syl 17 . . . . . . 7 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ran 𝐹𝑌)
101100, 93ssind 3795 . . . . . 6 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → ran 𝐹 ⊆ (𝑌 𝐽))
102 df-f 5791 . . . . . 6 (𝐹:𝑍⟶(𝑌 𝐽) ↔ (𝐹 Fn 𝑍 ∧ ran 𝐹 ⊆ (𝑌 𝐽)))
10392, 101, 102sylanbrc 694 . . . . 5 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → 𝐹:𝑍⟶(𝑌 𝐽))
10498, 49, 90, 103, 96lmbrf 20813 . . . 4 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝐹(⇝𝑡𝐾)𝑃 ↔ (𝑃 ∈ (𝑌 𝐽) ∧ ∀𝑣𝐾 (𝑃𝑣 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ∈ 𝑣))))
10587, 97, 1043bitr4d 298 . . 3 ((𝜑 ∧ (𝑃 𝐽 ∧ ran 𝐹 𝐽)) → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃))
106105ex 448 . 2 (𝜑 → ((𝑃 𝐽 ∧ ran 𝐹 𝐽) → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃)))
10714, 32, 106pm5.21ndd 367 1 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹(⇝𝑡𝐾)𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2892  wrex 2893  cin 3535  wss 3536   cuni 4363   class class class wbr 4574   × cxp 5023  ran crn 5026   Fn wfn 5782  wf 5783  cfv 5787  (class class class)co 6524  cc 9787  cz 11207  cuz 11516  t crest 15847  Topctop 20456  TopOnctopon 20457  𝑡clm 20779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-pre-lttri 9863  ax-pre-lttrn 9864
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-oadd 7425  df-er 7603  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-fi 8174  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-neg 10117  df-z 11208  df-uz 11517  df-rest 15849  df-topgen 15870  df-top 20460  df-bases 20461  df-topon 20462  df-lm 20782
This theorem is referenced by:  1stckgen  21106  minvecolem4b  26921  minvecolem4  26923  hhsscms  27323  lmlim  29124  climreeq  38481
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