Theorem cnpco 22326

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 22301	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
2	1	adantr 480	. . 3 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝐽 ∈ Top)
3		cnptop2 22302	. . . 4 ⊢ (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)) → 𝐿 ∈ Top)
4	3	adantl 481	. . 3 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝐿 ∈ Top)
5		eqid 2738	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	cnprcl 22304	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ ∪ 𝐽)
7	6	adantr 480	. . 3 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝑃 ∈ ∪ 𝐽)
8	2, 4, 7	3jca 1126	. 2 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → (𝐽 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝑃 ∈ ∪ 𝐽))
9		eqid 2738	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
10		eqid 2738	. . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿
11	9, 10	cnpf 22306	. . . . 5 ⊢ (𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃)) → 𝐺:∪ 𝐾⟶∪ 𝐿)
12	11	adantl 481	. . . 4 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝐺:∪ 𝐾⟶∪ 𝐿)
13	5, 9	cnpf 22306	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:∪ 𝐽⟶∪ 𝐾)
14	13	adantr 480	. . . 4 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → 𝐹:∪ 𝐽⟶∪ 𝐾)
15		fco 6608	. . . 4 ⊢ ((𝐺:∪ 𝐾⟶∪ 𝐿 ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿)
16	12, 14, 15	syl2anc 583	. . 3 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿)
17		simplr 765	. . . . . . 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 767	. . . . . . 7 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → 𝑧 ∈ 𝐿)
19		fvco3 6849	. . . . . . . . . 10 ⊢ ((𝐹:∪ 𝐽⟶∪ 𝐾 ∧ 𝑃 ∈ ∪ 𝐽) → ((𝐺 ∘ 𝐹)‘𝑃) = (𝐺‘(𝐹‘𝑃)))
20	14, 7, 19	syl2anc 583	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → ((𝐺 ∘ 𝐹)‘𝑃) = (𝐺‘(𝐹‘𝑃)))
21	20	adantr 480	. . . . . . . 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 769	. . . . . . . 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 2840	. . . . . . 7 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → (𝐺‘(𝐹‘𝑃)) ∈ 𝑧)
24		cnpimaex 22315	. . . . . . 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 1369	. . . . . 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 771	. . . . . . . 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 767	. . . . . . . 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 777	. . . . . . . 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 22315	. . . . . . . 8 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
30	26, 27, 28, 29	syl3anc 1369	. . . . . . 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 6144	. . . . . . . . . . 11 ⊢ ((𝐺 ∘ 𝐹) “ 𝑥) = (𝐺 “ (𝐹 “ 𝑥))
32		imass2 5999	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐺 “ (𝐹 “ 𝑥)) ⊆ (𝐺 “ 𝑦))
33	31, 32	eqsstrid 3965	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ (𝐺 “ 𝑦))
34		simprrr 778	. . . . . . . . . 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 3924	. . . . . . . . . 10 ⊢ (((𝐺 ∘ 𝐹) “ 𝑥) ⊆ (𝐺 “ 𝑦) → ((𝐺 “ 𝑦) ⊆ 𝑧 → ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ 𝑧))
36	33, 34, 35	syl2imc 41	. . . . . . . . 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 611	. . . . . . . 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 3201	. . . . . . 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 15	. . . . . 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 3219	. . . . 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 456	. . . 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 3107	. . 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 511	. 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 22298	. 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 582	1 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → (𝐺 ∘ 𝐹) ∈ ((𝐽 CnP 𝐿)‘𝑃))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ∪ cuni 4836   “ cima 5583   ∘ ccom 5584  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Topctop 21950   CnP ccnp 22284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-top 21951  df-topon 21968  df-cnp 22287

This theorem is referenced by:  limccnp  24960  limccnp2  24961  efrlim  26024