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

Theorem cncnp 20994
Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by NM, 15-May-2007.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cncnp ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝑋   𝑥,𝑌

Proof of Theorem cncnp
Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscn 20949 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
21simprbda 652 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
3 eqid 2621 . . . . . . 7 𝐽 = 𝐽
43cncnpi 20992 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
54ralrimiva 2960 . . . . 5 (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑥 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
65adantl 482 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
7 toponuni 20642 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
87ad2antrr 761 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝑋 = 𝐽)
98raleqdv 3133 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑥 𝐽𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
106, 9mpbird 247 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
112, 10jca 554 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
12 simprl 793 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹:𝑋𝑌)
13 cnvimass 5444 . . . . . . . . . 10 (𝐹𝑦) ⊆ dom 𝐹
14 fdm 6008 . . . . . . . . . . 11 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1514adantl 482 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → dom 𝐹 = 𝑋)
1613, 15syl5sseq 3632 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → (𝐹𝑦) ⊆ 𝑋)
17 ssralv 3645 . . . . . . . . 9 ((𝐹𝑦) ⊆ 𝑋 → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (𝐹𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
1816, 17syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (𝐹𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))
19 simprr 795 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
20 simpllr 798 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑦𝐾)
21 ffn 6002 . . . . . . . . . . . . . 14 (𝐹:𝑋𝑌𝐹 Fn 𝑋)
2221ad2antlr 762 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 Fn 𝑋)
23 simprl 793 . . . . . . . . . . . . 13 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝑥 ∈ (𝐹𝑦))
24 elpreima 6293 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑋 → (𝑥 ∈ (𝐹𝑦) ↔ (𝑥𝑋 ∧ (𝐹𝑥) ∈ 𝑦)))
2524simplbda 653 . . . . . . . . . . . . 13 ((𝐹 Fn 𝑋𝑥 ∈ (𝐹𝑦)) → (𝐹𝑥) ∈ 𝑦)
2622, 23, 25syl2anc 692 . . . . . . . . . . . 12 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (𝐹𝑥) ∈ 𝑦)
27 cnpimaex 20970 . . . . . . . . . . . 12 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ∧ 𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
2819, 20, 26, 27syl3anc 1323 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
29 simpllr 798 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → 𝐹:𝑋𝑌)
30 ffun 6005 . . . . . . . . . . . . . . 15 (𝐹:𝑋𝑌 → Fun 𝐹)
3129, 30syl 17 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → Fun 𝐹)
32 simp-4l 805 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐽 ∈ (TopOn‘𝑋))
33 toponss 20644 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢𝐽) → 𝑢𝑋)
3432, 33sylan 488 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → 𝑢𝑋)
3529, 14syl 17 . . . . . . . . . . . . . . 15 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → dom 𝐹 = 𝑋)
3634, 35sseqtr4d 3621 . . . . . . . . . . . . . 14 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → 𝑢 ⊆ dom 𝐹)
37 funimass3 6289 . . . . . . . . . . . . . 14 ((Fun 𝐹𝑢 ⊆ dom 𝐹) → ((𝐹𝑢) ⊆ 𝑦𝑢 ⊆ (𝐹𝑦)))
3831, 36, 37syl2anc 692 . . . . . . . . . . . . 13 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → ((𝐹𝑢) ⊆ 𝑦𝑢 ⊆ (𝐹𝑦)))
3938anbi2d 739 . . . . . . . . . . . 12 ((((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑢𝐽) → ((𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦) ↔ (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4039rexbidva 3042 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦) ↔ ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4128, 40mpbid 222 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ (𝑥 ∈ (𝐹𝑦) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦)))
4241expr 642 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) ∧ 𝑥 ∈ (𝐹𝑦)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4342ralimdva 2956 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → (∀𝑥 ∈ (𝐹𝑦)𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4418, 43syld 47 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) → ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
4544impr 648 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑦𝐾) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦)))
4645an32s 845 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦)))
47 topontop 20641 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4847ad3antrrr 765 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → 𝐽 ∈ Top)
49 eltop2 20690 . . . . . 6 (𝐽 ∈ Top → ((𝐹𝑦) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
5048, 49syl 17 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → ((𝐹𝑦) ∈ 𝐽 ↔ ∀𝑥 ∈ (𝐹𝑦)∃𝑢𝐽 (𝑥𝑢𝑢 ⊆ (𝐹𝑦))))
5146, 50mpbird 247 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
5251ralrimiva 2960 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)
531adantr 481 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
5412, 52, 53mpbir2and 956 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))) → 𝐹 ∈ (𝐽 Cn 𝐾))
5511, 54impbida 876 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3555   cuni 4402  ccnv 5073  dom cdm 5074  cima 5077  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Topctop 20617  TopOnctopon 20618   Cn ccn 20938   CnP ccnp 20939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-topgen 16025  df-top 20621  df-topon 20623  df-cn 20941  df-cnp 20942
This theorem is referenced by:  cncnp2  20995  cnnei  20996  cnconst2  20997  1stccn  21176  ptcn  21340  cnflf  21716  cnfcf  21756  symgtgp  21815  ghmcnp  21828  metcn  22258  txmetcn  22263  cnlimc  23558  dvcn  23590  dvcnvre  23686  psercn  24084  abelth  24099  cxpcn3  24389  cvmlift2lem11  31000  cvmlift2lem12  31001  cvmlift3lem8  31013  ioccncflimc  39399  cncfuni  39400  icccncfext  39401  icocncflimc  39403  cncfiooicclem1  39407  dirkercncflem2  39625  dirkercncflem4  39627  dirkercncf  39628  fourierdlem32  39660  fourierdlem33  39661  fourierdlem62  39689  fourierdlem93  39720  fourierdlem101  39728
  Copyright terms: Public domain W3C validator