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Mirrors > Home > ILE Home > Th. List > txcnmpt | Unicode version |
Description: A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
txcnmpt.1 | |
txcnmpt.2 |
Ref | Expression |
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txcnmpt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | txcnmpt.1 | . . . . . . 7 | |
2 | eqid 2139 | . . . . . . 7 | |
3 | 1, 2 | cnf 12373 | . . . . . 6 |
4 | 3 | adantr 274 | . . . . 5 |
5 | 4 | ffvelrnda 5555 | . . . 4 |
6 | eqid 2139 | . . . . . . 7 | |
7 | 1, 6 | cnf 12373 | . . . . . 6 |
8 | 7 | adantl 275 | . . . . 5 |
9 | 8 | ffvelrnda 5555 | . . . 4 |
10 | 5, 9 | opelxpd 4572 | . . 3 |
11 | txcnmpt.2 | . . 3 | |
12 | 10, 11 | fmptd 5574 | . 2 |
13 | 11 | mptpreima 5032 | . . . . . 6 |
14 | 4 | adantr 274 | . . . . . . . . . . . . 13 |
15 | 14 | adantr 274 | . . . . . . . . . . . 12 |
16 | ffn 5272 | . . . . . . . . . . . 12 | |
17 | elpreima 5539 | . . . . . . . . . . . 12 | |
18 | 15, 16, 17 | 3syl 17 | . . . . . . . . . . 11 |
19 | ibar 299 | . . . . . . . . . . . 12 | |
20 | 19 | adantl 275 | . . . . . . . . . . 11 |
21 | 18, 20 | bitr4d 190 | . . . . . . . . . 10 |
22 | 8 | ad2antrr 479 | . . . . . . . . . . . 12 |
23 | ffn 5272 | . . . . . . . . . . . 12 | |
24 | elpreima 5539 | . . . . . . . . . . . 12 | |
25 | 22, 23, 24 | 3syl 17 | . . . . . . . . . . 11 |
26 | ibar 299 | . . . . . . . . . . . 12 | |
27 | 26 | adantl 275 | . . . . . . . . . . 11 |
28 | 25, 27 | bitr4d 190 | . . . . . . . . . 10 |
29 | 21, 28 | anbi12d 464 | . . . . . . . . 9 |
30 | elin 3259 | . . . . . . . . 9 | |
31 | opelxp 4569 | . . . . . . . . 9 | |
32 | 29, 30, 31 | 3bitr4g 222 | . . . . . . . 8 |
33 | 32 | rabbi2dva 3284 | . . . . . . 7 |
34 | inss1 3296 | . . . . . . . . . 10 | |
35 | cnvimass 4902 | . . . . . . . . . 10 | |
36 | 34, 35 | sstri 3106 | . . . . . . . . 9 |
37 | 36, 14 | fssdm 5287 | . . . . . . . 8 |
38 | sseqin2 3295 | . . . . . . . 8 | |
39 | 37, 38 | sylib 121 | . . . . . . 7 |
40 | 33, 39 | eqtr3d 2174 | . . . . . 6 |
41 | 13, 40 | syl5eq 2184 | . . . . 5 |
42 | cntop1 12370 | . . . . . . . 8 | |
43 | 42 | adantl 275 | . . . . . . 7 |
44 | 43 | adantr 274 | . . . . . 6 |
45 | cnima 12389 | . . . . . . 7 | |
46 | 45 | ad2ant2r 500 | . . . . . 6 |
47 | cnima 12389 | . . . . . . 7 | |
48 | 47 | ad2ant2l 499 | . . . . . 6 |
49 | inopn 12170 | . . . . . 6 | |
50 | 44, 46, 48, 49 | syl3anc 1216 | . . . . 5 |
51 | 41, 50 | eqeltrd 2216 | . . . 4 |
52 | 51 | ralrimivva 2514 | . . 3 |
53 | vex 2689 | . . . . . 6 | |
54 | vex 2689 | . . . . . 6 | |
55 | 53, 54 | xpex 4654 | . . . . 5 |
56 | 55 | rgen2w 2488 | . . . 4 |
57 | eqid 2139 | . . . . 5 | |
58 | imaeq2 4877 | . . . . . 6 | |
59 | 58 | eleq1d 2208 | . . . . 5 |
60 | 57, 59 | ralrnmpo 5885 | . . . 4 |
61 | 56, 60 | ax-mp 5 | . . 3 |
62 | 52, 61 | sylibr 133 | . 2 |
63 | 1 | toptopon 12185 | . . . 4 TopOn |
64 | 43, 63 | sylib 121 | . . 3 TopOn |
65 | cntop2 12371 | . . . 4 | |
66 | cntop2 12371 | . . . 4 | |
67 | eqid 2139 | . . . . 5 | |
68 | 67 | txval 12424 | . . . 4 |
69 | 65, 66, 68 | syl2an 287 | . . 3 |
70 | toptopon2 12186 | . . . . 5 TopOn | |
71 | 65, 70 | sylib 121 | . . . 4 TopOn |
72 | toptopon2 12186 | . . . . 5 TopOn | |
73 | 66, 72 | sylib 121 | . . . 4 TopOn |
74 | txtopon 12431 | . . . 4 TopOn TopOn TopOn | |
75 | 71, 73, 74 | syl2an 287 | . . 3 TopOn |
76 | 64, 69, 75 | tgcn 12377 | . 2 |
77 | 12, 62, 76 | mpbir2and 928 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 crab 2420 cvv 2686 cin 3070 wss 3071 cop 3530 cuni 3736 cmpt 3989 cxp 4537 ccnv 4538 cdm 4539 crn 4540 cima 4542 wfn 5118 wf 5119 cfv 5123 (class class class)co 5774 cmpo 5776 ctg 12135 ctop 12164 TopOnctopon 12177 ccn 12354 ctx 12421 |
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-nul 3364 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-topgen 12141 df-top 12165 df-topon 12178 df-bases 12210 df-cn 12357 df-tx 12422 |
This theorem is referenced by: uptx 12443 cnmpt1t 12454 cnmpt2t 12462 |
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