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Theorem cnpco 23262
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 23237 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
21adantr 479 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐽 ∈ Top)
3 cnptop2 23238 . . . 4 (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) → 𝐿 ∈ Top)
43adantl 480 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐿 ∈ Top)
5 eqid 2726 . . . . 5 𝐽 = 𝐽
65cnprcl 23240 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 𝐽)
76adantr 479 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝑃 𝐽)
82, 4, 73jca 1125 . 2 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → (𝐽 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝑃 𝐽))
9 eqid 2726 . . . . . 6 𝐾 = 𝐾
10 eqid 2726 . . . . . 6 𝐿 = 𝐿
119, 10cnpf 23242 . . . . 5 (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) → 𝐺: 𝐾 𝐿)
1211adantl 480 . . . 4 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐺: 𝐾 𝐿)
135, 9cnpf 23242 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹: 𝐽 𝐾)
1413adantr 479 . . . 4 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐹: 𝐽 𝐾)
15 fco 6752 . . . 4 ((𝐺: 𝐾 𝐿𝐹: 𝐽 𝐾) → (𝐺𝐹): 𝐽 𝐿)
1612, 14, 15syl2anc 582 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → (𝐺𝐹): 𝐽 𝐿)
17 simplr 767 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)))
18 simprl 769 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → 𝑧𝐿)
19 fvco3 7001 . . . . . . . . . 10 ((𝐹: 𝐽 𝐾𝑃 𝐽) → ((𝐺𝐹)‘𝑃) = (𝐺‘(𝐹𝑃)))
2014, 7, 19syl2anc 582 . . . . . . . . 9 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → ((𝐺𝐹)‘𝑃) = (𝐺‘(𝐹𝑃)))
2120adantr 479 . . . . . . . 8 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺𝐹)‘𝑃) = (𝐺‘(𝐹𝑃)))
22 simprr 771 . . . . . . . 8 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺𝐹)‘𝑃) ∈ 𝑧)
2321, 22eqeltrrd 2827 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → (𝐺‘(𝐹𝑃)) ∈ 𝑧)
24 cnpimaex 23251 . . . . . . 7 ((𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) ∧ 𝑧𝐿 ∧ (𝐺‘(𝐹𝑃)) ∈ 𝑧) → ∃𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))
2517, 18, 23, 24syl3anc 1368 . . . . . 6 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))
26 simplll 773 . . . . . . . 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 769 . . . . . . . 8 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → 𝑦𝐾)
28 simprrl 779 . . . . . . . 8 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → (𝐹𝑃) ∈ 𝑦)
29 cnpimaex 23251 . . . . . . . 8 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
3026, 27, 28, 29syl3anc 1368 . . . . . . 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 6262 . . . . . . . . . . 11 ((𝐺𝐹) “ 𝑥) = (𝐺 “ (𝐹𝑥))
32 imass2 6112 . . . . . . . . . . 11 ((𝐹𝑥) ⊆ 𝑦 → (𝐺 “ (𝐹𝑥)) ⊆ (𝐺𝑦))
3331, 32eqsstrid 4028 . . . . . . . . . 10 ((𝐹𝑥) ⊆ 𝑦 → ((𝐺𝐹) “ 𝑥) ⊆ (𝐺𝑦))
34 simprrr 780 . . . . . . . . . 10 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → (𝐺𝑦) ⊆ 𝑧)
35 sstr2 3986 . . . . . . . . . 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 610 . . . . . . . 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 3160 . . . . . . 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 3151 . . . . 5 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧))
4140expr 455 . . . 4 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ 𝑧𝐿) → (((𝐺𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))
4241ralrimiva 3136 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → ∀𝑧𝐿 (((𝐺𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))
4316, 42jca 510 . 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 23234 . 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 581 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 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wrex 3060  wss 3947   cuni 4913  cima 5685  ccom 5686  wf 6550  cfv 6554  (class class class)co 7424  Topctop 22886   CnP ccnp 23220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-map 8857  df-top 22887  df-topon 22904  df-cnp 23223
This theorem is referenced by:  limccnp  25911  limccnp2  25912  efrlim  26997  efrlimOLD  26998
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