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Theorem cnpco 23290
Description: The composition of a function 𝐹 continuous at 𝑃 with a function continuous at (𝐹𝑃) is continuous at 𝑃. Proposition 2 of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
cnpco ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → (𝐺𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))

Proof of Theorem cnpco
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnptop1 23265 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
21adantr 480 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐽 ∈ Top)
3 cnptop2 23266 . . . 4 (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) → 𝐿 ∈ Top)
43adantl 481 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐿 ∈ Top)
5 eqid 2734 . . . . 5 𝐽 = 𝐽
65cnprcl 23268 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 𝐽)
76adantr 480 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝑃 𝐽)
82, 4, 73jca 1127 . 2 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → (𝐽 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝑃 𝐽))
9 eqid 2734 . . . . . 6 𝐾 = 𝐾
10 eqid 2734 . . . . . 6 𝐿 = 𝐿
119, 10cnpf 23270 . . . . 5 (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) → 𝐺: 𝐾 𝐿)
1211adantl 481 . . . 4 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐺: 𝐾 𝐿)
135, 9cnpf 23270 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹: 𝐽 𝐾)
1413adantr 480 . . . 4 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐹: 𝐽 𝐾)
15 fco 6760 . . . 4 ((𝐺: 𝐾 𝐿𝐹: 𝐽 𝐾) → (𝐺𝐹): 𝐽 𝐿)
1612, 14, 15syl2anc 584 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → (𝐺𝐹): 𝐽 𝐿)
17 simplr 769 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)))
18 simprl 771 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → 𝑧𝐿)
19 fvco3 7007 . . . . . . . . . 10 ((𝐹: 𝐽 𝐾𝑃 𝐽) → ((𝐺𝐹)‘𝑃) = (𝐺‘(𝐹𝑃)))
2014, 7, 19syl2anc 584 . . . . . . . . 9 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → ((𝐺𝐹)‘𝑃) = (𝐺‘(𝐹𝑃)))
2120adantr 480 . . . . . . . 8 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺𝐹)‘𝑃) = (𝐺‘(𝐹𝑃)))
22 simprr 773 . . . . . . . 8 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺𝐹)‘𝑃) ∈ 𝑧)
2321, 22eqeltrrd 2839 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → (𝐺‘(𝐹𝑃)) ∈ 𝑧)
24 cnpimaex 23279 . . . . . . 7 ((𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) ∧ 𝑧𝐿 ∧ (𝐺‘(𝐹𝑃)) ∈ 𝑧) → ∃𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))
2517, 18, 23, 24syl3anc 1370 . . . . . 6 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))
26 simplll 775 . . . . . . . 8 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
27 simprl 771 . . . . . . . 8 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → 𝑦𝐾)
28 simprrl 781 . . . . . . . 8 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → (𝐹𝑃) ∈ 𝑦)
29 cnpimaex 23279 . . . . . . . 8 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
3026, 27, 28, 29syl3anc 1370 . . . . . . 7 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
31 imaco 6272 . . . . . . . . . . 11 ((𝐺𝐹) “ 𝑥) = (𝐺 “ (𝐹𝑥))
32 imass2 6122 . . . . . . . . . . 11 ((𝐹𝑥) ⊆ 𝑦 → (𝐺 “ (𝐹𝑥)) ⊆ (𝐺𝑦))
3331, 32eqsstrid 4043 . . . . . . . . . 10 ((𝐹𝑥) ⊆ 𝑦 → ((𝐺𝐹) “ 𝑥) ⊆ (𝐺𝑦))
34 simprrr 782 . . . . . . . . . 10 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → (𝐺𝑦) ⊆ 𝑧)
35 sstr2 4001 . . . . . . . . . 10 (((𝐺𝐹) “ 𝑥) ⊆ (𝐺𝑦) → ((𝐺𝑦) ⊆ 𝑧 → ((𝐺𝐹) “ 𝑥) ⊆ 𝑧))
3633, 34, 35syl2imc 41 . . . . . . . . 9 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → ((𝐹𝑥) ⊆ 𝑦 → ((𝐺𝐹) “ 𝑥) ⊆ 𝑧))
3736anim2d 612 . . . . . . . 8 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → ((𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))
3837reximdv 3167 . . . . . . 7 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → (∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))
3930, 38mpd 15 . . . . . 6 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧))
4025, 39rexlimddv 3158 . . . . 5 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧))
4140expr 456 . . . 4 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ 𝑧𝐿) → (((𝐺𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))
4241ralrimiva 3143 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → ∀𝑧𝐿 (((𝐺𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))
4316, 42jca 511 . 2 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → ((𝐺𝐹): 𝐽 𝐿 ∧ ∀𝑧𝐿 (((𝐺𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧))))
445, 10iscnp2 23262 . 2 ((𝐺𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝑃 𝐽) ∧ ((𝐺𝐹): 𝐽 𝐿 ∧ ∀𝑧𝐿 (((𝐺𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))))
458, 43, 44sylanbrc 583 1 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → (𝐺𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  wss 3962   cuni 4911  cima 5691  ccom 5692  wf 6558  cfv 6562  (class class class)co 7430  Topctop 22914   CnP ccnp 23248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-top 22915  df-topon 22932  df-cnp 23251
This theorem is referenced by:  limccnp  25940  limccnp2  25941  efrlim  27026  efrlimOLD  27027
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