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Theorem cnpco 22489
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 22464 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
21adantr 481 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐽 ∈ Top)
3 cnptop2 22465 . . . 4 (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) → 𝐿 ∈ Top)
43adantl 482 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐿 ∈ Top)
5 eqid 2737 . . . . 5 𝐽 = 𝐽
65cnprcl 22467 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 𝐽)
76adantr 481 . . 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 2737 . . . . . 6 𝐾 = 𝐾
10 eqid 2737 . . . . . 6 𝐿 = 𝐿
119, 10cnpf 22469 . . . . 5 (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) → 𝐺: 𝐾 𝐿)
1211adantl 482 . . . 4 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐺: 𝐾 𝐿)
135, 9cnpf 22469 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹: 𝐽 𝐾)
1413adantr 481 . . . 4 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐹: 𝐽 𝐾)
15 fco 6659 . . . 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 766 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)))
18 simprl 768 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → 𝑧𝐿)
19 fvco3 6904 . . . . . . . . . 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 481 . . . . . . . 8 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺𝐹)‘𝑃) = (𝐺‘(𝐹𝑃)))
22 simprr 770 . . . . . . . 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 22478 . . . . . . 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 772 . . . . . . . 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 768 . . . . . . . 8 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → 𝑦𝐾)
28 simprrl 778 . . . . . . . 8 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → (𝐹𝑃) ∈ 𝑦)
29 cnpimaex 22478 . . . . . . . 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 6174 . . . . . . . . . . 11 ((𝐺𝐹) “ 𝑥) = (𝐺 “ (𝐹𝑥))
32 imass2 6025 . . . . . . . . . . 11 ((𝐹𝑥) ⊆ 𝑦 → (𝐺 “ (𝐹𝑥)) ⊆ (𝐺𝑦))
3331, 32eqsstrid 3978 . . . . . . . . . 10 ((𝐹𝑥) ⊆ 𝑦 → ((𝐺𝐹) “ 𝑥) ⊆ (𝐺𝑦))
34 simprrr 779 . . . . . . . . . 10 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → (𝐺𝑦) ⊆ 𝑧)
35 sstr2 3937 . . . . . . . . . 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 3164 . . . . . . 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 3155 . . . . 5 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧))
4140expr 457 . . . 4 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ 𝑧𝐿) → (((𝐺𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))
4241ralrimiva 3140 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → ∀𝑧𝐿 (((𝐺𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))
4316, 42jca 512 . 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 22461 . 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 396  w3a 1086   = wceq 1540  wcel 2105  wral 3062  wrex 3071  wss 3896   cuni 4848  cima 5608  ccom 5609  wf 6459  cfv 6463  (class class class)co 7313  Topctop 22113   CnP ccnp 22447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-1st 7874  df-2nd 7875  df-map 8663  df-top 22114  df-topon 22131  df-cnp 22450
This theorem is referenced by:  limccnp  25126  limccnp2  25127  efrlim  26190
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