MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnpco Structured version   Visualization version   GIF version

Theorem cnpco 21273
Description: The composition of two continuous functions at point 𝑃 is a continuous function at point 𝑃. Proposition 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 21248 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
21adantr 472 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐽 ∈ Top)
3 cnptop2 21249 . . . 4 (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) → 𝐿 ∈ Top)
43adantl 473 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐿 ∈ Top)
5 eqid 2760 . . . . 5 𝐽 = 𝐽
65cnprcl 21251 . . . 4 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 𝐽)
76adantr 472 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝑃 𝐽)
82, 4, 73jca 1123 . 2 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → (𝐽 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝑃 𝐽))
9 eqid 2760 . . . . . 6 𝐾 = 𝐾
10 eqid 2760 . . . . . 6 𝐿 = 𝐿
119, 10cnpf 21253 . . . . 5 (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) → 𝐺: 𝐾 𝐿)
1211adantl 473 . . . 4 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐺: 𝐾 𝐿)
135, 9cnpf 21253 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹: 𝐽 𝐾)
1413adantr 472 . . . 4 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → 𝐹: 𝐽 𝐾)
15 fco 6219 . . . 4 ((𝐺: 𝐾 𝐿𝐹: 𝐽 𝐾) → (𝐺𝐹): 𝐽 𝐿)
1612, 14, 15syl2anc 696 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → (𝐺𝐹): 𝐽 𝐿)
17 simplr 809 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)))
18 simprl 811 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → 𝑧𝐿)
19 fvco3 6437 . . . . . . . . . 10 ((𝐹: 𝐽 𝐾𝑃 𝐽) → ((𝐺𝐹)‘𝑃) = (𝐺‘(𝐹𝑃)))
2014, 7, 19syl2anc 696 . . . . . . . . 9 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → ((𝐺𝐹)‘𝑃) = (𝐺‘(𝐹𝑃)))
2120adantr 472 . . . . . . . 8 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺𝐹)‘𝑃) = (𝐺‘(𝐹𝑃)))
22 simprr 813 . . . . . . . 8 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺𝐹)‘𝑃) ∈ 𝑧)
2321, 22eqeltrrd 2840 . . . . . . 7 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → (𝐺‘(𝐹𝑃)) ∈ 𝑧)
24 cnpimaex 21262 . . . . . . 7 ((𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃)) ∧ 𝑧𝐿 ∧ (𝐺‘(𝐹𝑃)) ∈ 𝑧) → ∃𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))
2517, 18, 23, 24syl3anc 1477 . . . . . 6 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑦𝐾 ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))
26 simplll 815 . . . . . . . 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 811 . . . . . . . 8 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → 𝑦𝐾)
28 simprrl 823 . . . . . . . 8 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → (𝐹𝑃) ∈ 𝑦)
29 cnpimaex 21262 . . . . . . . 8 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦𝐾 ∧ (𝐹𝑃) ∈ 𝑦) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝐹𝑥) ⊆ 𝑦))
3026, 27, 28, 29syl3anc 1477 . . . . . . 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 5801 . . . . . . . . . . 11 ((𝐺𝐹) “ 𝑥) = (𝐺 “ (𝐹𝑥))
32 imass2 5659 . . . . . . . . . . 11 ((𝐹𝑥) ⊆ 𝑦 → (𝐺 “ (𝐹𝑥)) ⊆ (𝐺𝑦))
3331, 32syl5eqss 3790 . . . . . . . . . 10 ((𝐹𝑥) ⊆ 𝑦 → ((𝐺𝐹) “ 𝑥) ⊆ (𝐺𝑦))
34 simprrr 824 . . . . . . . . . 10 ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦𝐾 ∧ ((𝐹𝑃) ∈ 𝑦 ∧ (𝐺𝑦) ⊆ 𝑧))) → (𝐺𝑦) ⊆ 𝑧)
35 sstr2 3751 . . . . . . . . . 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 590 . . . . . . . 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 3154 . . . . . . 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 3173 . . . . 5 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ (𝑧𝐿 ∧ ((𝐺𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧))
4140expr 644 . . . 4 (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) ∧ 𝑧𝐿) → (((𝐺𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))
4241ralrimiva 3104 . . 3 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹𝑃))) → ∀𝑧𝐿 (((𝐺𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥𝐽 (𝑃𝑥 ∧ ((𝐺𝐹) “ 𝑥) ⊆ 𝑧)))
4316, 42jca 555 . 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 21245 . 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 701 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 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  wss 3715   cuni 4588  cima 5269  ccom 5270  wf 6045  cfv 6049  (class class class)co 6813  Topctop 20900   CnP ccnp 21231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-top 20901  df-topon 20918  df-cnp 21234
This theorem is referenced by:  limccnp  23854  limccnp2  23855  efrlim  24895
  Copyright terms: Public domain W3C validator