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Mirrors > Home > ILE Home > Th. List > cncnp2m | Unicode version |
Description: A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.) |
Ref | Expression |
---|---|
cncnp.1 |
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cncnp.2 |
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Ref | Expression |
---|---|
cncnp2m |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntop1 12209 |
. . . . 5
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2 | cncnp.1 |
. . . . . 6
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3 | 2 | toptopon 12025 |
. . . . 5
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4 | 1, 3 | sylib 121 |
. . . 4
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5 | cntop2 12210 |
. . . . 5
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6 | cncnp.2 |
. . . . . 6
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7 | 6 | toptopon 12025 |
. . . . 5
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8 | 5, 7 | sylib 121 |
. . . 4
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9 | 2, 6 | cnf 12212 |
. . . 4
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10 | 4, 8, 9 | jca31 305 |
. . 3
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11 | 10 | adantl 273 |
. 2
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12 | 3 | biimpi 119 |
. . . . 5
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13 | 12 | 3ad2ant1 985 |
. . . 4
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14 | 13 | adantr 272 |
. . 3
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15 | 7 | biimpi 119 |
. . . . 5
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16 | 15 | 3ad2ant2 986 |
. . . 4
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17 | 16 | adantr 272 |
. . 3
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18 | r19.2m 3415 |
. . . . . . 7
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19 | 18 | ex 114 |
. . . . . 6
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20 | 19 | 3ad2ant3 987 |
. . . . 5
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21 | cnpf2 12215 |
. . . . . . . 8
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22 | 21 | 3expia 1166 |
. . . . . . 7
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23 | 22 | rexlimdvw 2527 |
. . . . . 6
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24 | 13, 16, 23 | syl2anc 406 |
. . . . 5
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25 | 20, 24 | syld 45 |
. . . 4
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26 | 25 | imp 123 |
. . 3
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27 | 14, 17, 26 | jca31 305 |
. 2
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28 | cncnp 12238 |
. . 3
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29 | 28 | baibd 891 |
. 2
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30 | 11, 27, 29 | pm5.21nd 884 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-map 6498 df-topgen 11981 df-top 12005 df-topon 12018 df-cn 12197 df-cnp 12198 |
This theorem is referenced by: (None) |
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