Theorem cnpco 23478

Description: The composition of a function 𝐹 continuous at 𝑃 with a function continuous at (𝐹‘𝑃) 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)‘(𝐹‘𝑃))) → (𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))

Proof of Theorem cnpco

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnptop1 23453	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
2	1	adantr 486	. . 3 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝐽 ∈ Top)
3		cnptop2 23454	. . . 4 ⊢ (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)) → 𝐿 ∈ Top)
4	3	adantl 487	. . 3 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝐿 ∈ Top)
5		eqid 2765	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	cnprcl 23456	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ ∪ 𝐽)
7	6	adantr 486	. . 3 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝑃 ∈ ∪ 𝐽)
8	2, 4, 7	3jca 1146	. 2 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → (𝐽 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝑃 ∈ ∪ 𝐽))
9		eqid 2765	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
10		eqid 2765	. . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿
11	9, 10	cnpf 23458	. . . . 5 ⊢ (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)) → 𝐺:∪ 𝐾⟶∪ 𝐿)
12	11	adantl 487	. . . 4 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝐺:∪ 𝐾⟶∪ 𝐿)
13	5, 9	cnpf 23458	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:∪ 𝐽⟶∪ 𝐾)
14	13	adantr 486	. . . 4 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝐹:∪ 𝐽⟶∪ 𝐾)
15		fco 6734	. . . 4 ⊢ ((𝐺:∪ 𝐾⟶∪ 𝐿 ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿)
16	12, 14, 15	syl2anc 596	. . 3 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿)
17		simplr 781	. . . . . . 7 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)))
18		simprl 783	. . . . . . 7 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → 𝑧 ∈ 𝐿)
19		fvco3 6985	. . . . . . . . . 10 ⊢ ((𝐹:∪ 𝐽⟶∪ 𝐾 ∧ 𝑃 ∈ ∪ 𝐽) → ((𝐺 ∘ 𝐹)‘𝑃) = (𝐺‘(𝐹‘𝑃)))
20	14, 7, 19	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → ((𝐺 ∘ 𝐹)‘𝑃) = (𝐺‘(𝐹‘𝑃)))
21	20	adantr 486	. . . . . . . 8 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺 ∘ 𝐹)‘𝑃) = (𝐺‘(𝐹‘𝑃)))
22		simprr 785	. . . . . . . 8 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)
23	21, 22	eqeltrrd 2866	. . . . . . 7 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → (𝐺‘(𝐹‘𝑃)) ∈ 𝑧)
24		cnpimaex 23467	. . . . . . 7 ⊢ ((𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)) ∧ 𝑧 ∈ 𝐿 ∧ (𝐺‘(𝐹‘𝑃)) ∈ 𝑧) → ∃𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))
25	17, 18, 23, 24	syl3anc 1398	. . . . . 6 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑦 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))
26		simplll 787	. . . . . . . 8 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
27		simprl 783	. . . . . . . 8 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → 𝑦 ∈ 𝐾)
28		simprrl 793	. . . . . . . 8 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → (𝐹‘𝑃) ∈ 𝑦)
29		cnpimaex 23467	. . . . . . . 8 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
30	26, 27, 28, 29	syl3anc 1398	. . . . . . 7 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
31		imaco 6254	. . . . . . . . . . 11 ⊢ ((𝐺 ∘ 𝐹) “ 𝑥) = (𝐺 “ (𝐹 “ 𝑥))
32		imass2 6106	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐺 “ (𝐹 “ 𝑥)) ⊆ (𝐺 “ 𝑦))
33	31, 32	eqsstrid 3976	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ (𝐺 “ 𝑦))
34		simprrr 794	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → (𝐺 “ 𝑦) ⊆ 𝑧)
35		sstr2 3945	. . . . . . . . . 10 ⊢ (((𝐺 ∘ 𝐹) “ 𝑥) ⊆ (𝐺 “ 𝑦) → ((𝐺 “ 𝑦) ⊆ 𝑧 → ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))
36	33, 34, 35	syl2imc 42	. . . . . . . . 9 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → ((𝐹 “ 𝑥) ⊆ 𝑦 → ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))
37	36	anim2d 624	. . . . . . . 8 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))
38	37	reximdv 3182	. . . . . . 7 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))
39	30, 38	mpd 16	. . . . . 6 ⊢ ((((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) ∧ (𝑦 ∈ 𝐾 ∧ ((𝐹‘𝑃) ∈ 𝑦 ∧ (𝐺 “ 𝑦) ⊆ 𝑧))) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))
40	25, 39	rexlimddv 3174	. . . . 5 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))
41	40	expr 462	. . . 4 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ 𝑧 ∈ 𝐿) → (((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))
42	41	ralrimiva 3159	. . 3 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → ∀𝑧 ∈ 𝐿 (((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))
43	16, 42	jca 521	. 2 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → ((𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿 ∧ ∀𝑧 ∈ 𝐿 (((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))))
44	5, 10	iscnp2 23450	. 2 ⊢ ((𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝑃 ∈ ∪ 𝐽) ∧ ((𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿 ∧ ∀𝑧 ∈ 𝐿 (((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧)))))
45	8, 43, 44	sylanbrc 595	1 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → (𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   ⊆ wss 3906  ∪ cuni 4874   “ cima 5666   ∘ ccom 5667  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  Topctop 23104   CnP ccnp 23436

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-top 23105  df-topon 23122  df-cnp 23439

This theorem is used by:  limccnp  26105  limccnp2  26106  efrlim  27189