Theorem cnpco 23209

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 23184	. . . 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 23185	. . . 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 2734	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	cnprcl 23187	. . . 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 1128	. 2 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → (𝐽 ∈ Top ∧ 𝐿 ∈ Top ∧ 𝑃 ∈ ∪ 𝐽))
9		eqid 2734	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
10		eqid 2734	. . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿
11	9, 10	cnpf 23189	. . . . 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 23189	. . . . 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 6684	. . . 4 ⊢ ((𝐺:∪ 𝐾⟶∪ 𝐿 ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿)
16	12, 14, 15	syl2anc 584	. . 3 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) → (𝐺 ∘ 𝐹):∪ 𝐽⟶∪ 𝐿)
17		simplr 768	. . . . . . 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 770	. . . . . . 7 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → 𝑧 ∈ 𝐿)
19		fvco3 6931	. . . . . . . . . 10 ⊢ ((𝐹:∪ 𝐽⟶∪ 𝐾 ∧ 𝑃 ∈ ∪ 𝐽) → ((𝐺 ∘ 𝐹)‘𝑃) = (𝐺‘(𝐹‘𝑃)))
20	14, 7, 19	syl2anc 584	. . . . . . . . 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 772	. . . . . . . 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 2835	. . . . . . 7 ⊢ (((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐺 ∈ ((𝐾 CnP 𝐿)‘(𝐹‘𝑃))) ∧ (𝑧 ∈ 𝐿 ∧ ((𝐺 ∘ 𝐹)‘𝑃) ∈ 𝑧)) → (𝐺‘(𝐹‘𝑃)) ∈ 𝑧)
24		cnpimaex 23198	. . . . . . 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 1373	. . . . . 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 774	. . . . . . . 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 770	. . . . . . . 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 780	. . . . . . . 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 23198	. . . . . . . 8 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
30	26, 27, 28, 29	syl3anc 1373	. . . . . . 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 6207	. . . . . . . . . . 11 ⊢ ((𝐺 ∘ 𝐹) “ 𝑥) = (𝐺 “ (𝐹 “ 𝑥))
32		imass2 6059	. . . . . . . . . . 11 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐺 “ (𝐹 “ 𝑥)) ⊆ (𝐺 “ 𝑦))
33	31, 32	eqsstrid 3970	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → ((𝐺 ∘ 𝐹) “ 𝑥) ⊆ (𝐺 “ 𝑦))
34		simprrr 781	. . . . . . . . . 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 3938	. . . . . . . . . 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 612	. . . . . . . 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 3149	. . . . . . 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 3141	. . . . 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 3126	. . 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 23181	. 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 583	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 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058   ⊆ wss 3899  ∪ cuni 4861   “ cima 5625   ∘ ccom 5626  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  Topctop 22835   CnP ccnp 23167

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-top 22836  df-topon 22853  df-cnp 23170

This theorem is referenced by:  limccnp  25846  limccnp2  25847  efrlim  26933  efrlimOLD  26934