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Mirrors > Home > ILE Home > Th. List > cnptopco | Unicode version |
Description: The composition of a function continuous at with a function continuous at is continuous at . Proposition 2 of [BourbakiTop1] p. I.9. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
cnptopco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl2 985 | . . . . 5 | |
2 | toptopon2 12186 | . . . . 5 TopOn | |
3 | 1, 2 | sylib 121 | . . . 4 TopOn |
4 | simpl3 986 | . . . . 5 | |
5 | toptopon2 12186 | . . . . 5 TopOn | |
6 | 4, 5 | sylib 121 | . . . 4 TopOn |
7 | simprr 521 | . . . 4 | |
8 | cnpf2 12376 | . . . 4 TopOn TopOn | |
9 | 3, 6, 7, 8 | syl3anc 1216 | . . 3 |
10 | simpl1 984 | . . . . 5 | |
11 | toptopon2 12186 | . . . . 5 TopOn | |
12 | 10, 11 | sylib 121 | . . . 4 TopOn |
13 | simprl 520 | . . . 4 | |
14 | cnpf2 12376 | . . . 4 TopOn TopOn | |
15 | 12, 3, 13, 14 | syl3anc 1216 | . . 3 |
16 | fco 5288 | . . 3 | |
17 | 9, 15, 16 | syl2anc 408 | . 2 |
18 | 3 | adantr 274 | . . . . . 6 TopOn |
19 | 6 | adantr 274 | . . . . . 6 TopOn |
20 | cnprcl2k 12375 | . . . . . . . . 9 TopOn | |
21 | 12, 1, 13, 20 | syl3anc 1216 | . . . . . . . 8 |
22 | 15, 21 | ffvelrnd 5556 | . . . . . . 7 |
23 | 22 | adantr 274 | . . . . . 6 |
24 | 7 | adantr 274 | . . . . . 6 |
25 | simprl 520 | . . . . . 6 | |
26 | fvco3 5492 | . . . . . . . . 9 | |
27 | 15, 21, 26 | syl2anc 408 | . . . . . . . 8 |
28 | 27 | adantr 274 | . . . . . . 7 |
29 | simprr 521 | . . . . . . 7 | |
30 | 28, 29 | eqeltrrd 2217 | . . . . . 6 |
31 | icnpimaex 12380 | . . . . . 6 TopOn TopOn | |
32 | 18, 19, 23, 24, 25, 30, 31 | syl33anc 1231 | . . . . 5 |
33 | 12 | ad2antrr 479 | . . . . . . 7 TopOn |
34 | 3 | ad2antrr 479 | . . . . . . 7 TopOn |
35 | 21 | ad2antrr 479 | . . . . . . 7 |
36 | simplll 522 | . . . . . . . 8 | |
37 | 36 | adantlll 471 | . . . . . . 7 |
38 | simprl 520 | . . . . . . 7 | |
39 | simprrl 528 | . . . . . . 7 | |
40 | icnpimaex 12380 | . . . . . . 7 TopOn TopOn | |
41 | 33, 34, 35, 37, 38, 39, 40 | syl33anc 1231 | . . . . . 6 |
42 | imaco 5044 | . . . . . . . . . . 11 | |
43 | imass2 4915 | . . . . . . . . . . 11 | |
44 | 42, 43 | eqsstrid 3143 | . . . . . . . . . 10 |
45 | simprrr 529 | . . . . . . . . . 10 | |
46 | sstr2 3104 | . . . . . . . . . 10 | |
47 | 44, 45, 46 | syl2imc 39 | . . . . . . . . 9 |
48 | 47 | adantlll 471 | . . . . . . . 8 |
49 | 48 | anim2d 335 | . . . . . . 7 |
50 | 49 | reximdv 2533 | . . . . . 6 |
51 | 41, 50 | mpd 13 | . . . . 5 |
52 | 32, 51 | rexlimddv 2554 | . . . 4 |
53 | 52 | expr 372 | . . 3 |
54 | 53 | ralrimiva 2505 | . 2 |
55 | iscnp 12368 | . . 3 TopOn TopOn | |
56 | 12, 6, 21, 55 | syl3anc 1216 | . 2 |
57 | 17, 54, 56 | mpbir2and 928 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2416 wrex 2417 wss 3071 cuni 3736 cima 4542 ccom 4543 wf 5119 cfv 5123 (class class class)co 5774 ctop 12164 TopOnctopon 12177 ccnp 12355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-top 12165 df-topon 12178 df-cnp 12358 |
This theorem is referenced by: (None) |
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