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Mirrors > Home > ILE Home > Th. List > txhmeo | Unicode version |
Description: Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
txhmeo.1 | |
txhmeo.2 | |
txhmeo.3 | |
txhmeo.4 |
Ref | Expression |
---|---|
txhmeo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | txhmeo.3 | . . . . . 6 | |
2 | hmeocn 12474 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | cntop1 12370 | . . . . 5 | |
5 | 3, 4 | syl 14 | . . . 4 |
6 | txhmeo.1 | . . . . 5 | |
7 | 6 | toptopon 12185 | . . . 4 TopOn |
8 | 5, 7 | sylib 121 | . . 3 TopOn |
9 | txhmeo.4 | . . . . . 6 | |
10 | hmeocn 12474 | . . . . . 6 | |
11 | 9, 10 | syl 14 | . . . . 5 |
12 | cntop1 12370 | . . . . 5 | |
13 | 11, 12 | syl 14 | . . . 4 |
14 | txhmeo.2 | . . . . 5 | |
15 | 14 | toptopon 12185 | . . . 4 TopOn |
16 | 13, 15 | sylib 121 | . . 3 TopOn |
17 | 8, 16 | cnmpt1st 12457 | . . . 4 |
18 | 8, 16, 17, 3 | cnmpt21f 12461 | . . 3 |
19 | 8, 16 | cnmpt2nd 12458 | . . . 4 |
20 | 8, 16, 19, 11 | cnmpt21f 12461 | . . 3 |
21 | 8, 16, 18, 20 | cnmpt2t 12462 | . 2 |
22 | vex 2689 | . . . . . . . . . . 11 | |
23 | vex 2689 | . . . . . . . . . . 11 | |
24 | 22, 23 | op1std 6046 | . . . . . . . . . 10 |
25 | 24 | fveq2d 5425 | . . . . . . . . 9 |
26 | 22, 23 | op2ndd 6047 | . . . . . . . . . 10 |
27 | 26 | fveq2d 5425 | . . . . . . . . 9 |
28 | 25, 27 | opeq12d 3713 | . . . . . . . 8 |
29 | 28 | mpompt 5863 | . . . . . . 7 |
30 | 29 | eqcomi 2143 | . . . . . 6 |
31 | eqid 2139 | . . . . . . . . . 10 | |
32 | 6, 31 | cnf 12373 | . . . . . . . . 9 |
33 | 3, 32 | syl 14 | . . . . . . . 8 |
34 | xp1st 6063 | . . . . . . . 8 | |
35 | ffvelrn 5553 | . . . . . . . 8 | |
36 | 33, 34, 35 | syl2an 287 | . . . . . . 7 |
37 | eqid 2139 | . . . . . . . . . 10 | |
38 | 14, 37 | cnf 12373 | . . . . . . . . 9 |
39 | 11, 38 | syl 14 | . . . . . . . 8 |
40 | xp2nd 6064 | . . . . . . . 8 | |
41 | ffvelrn 5553 | . . . . . . . 8 | |
42 | 39, 40, 41 | syl2an 287 | . . . . . . 7 |
43 | 36, 42 | opelxpd 4572 | . . . . . 6 |
44 | 6, 31 | hmeof1o 12478 | . . . . . . . . . 10 |
45 | 1, 44 | syl 14 | . . . . . . . . 9 |
46 | f1ocnv 5380 | . . . . . . . . 9 | |
47 | f1of 5367 | . . . . . . . . 9 | |
48 | 45, 46, 47 | 3syl 17 | . . . . . . . 8 |
49 | xp1st 6063 | . . . . . . . 8 | |
50 | ffvelrn 5553 | . . . . . . . 8 | |
51 | 48, 49, 50 | syl2an 287 | . . . . . . 7 |
52 | 14, 37 | hmeof1o 12478 | . . . . . . . . . 10 |
53 | 9, 52 | syl 14 | . . . . . . . . 9 |
54 | f1ocnv 5380 | . . . . . . . . 9 | |
55 | f1of 5367 | . . . . . . . . 9 | |
56 | 53, 54, 55 | 3syl 17 | . . . . . . . 8 |
57 | xp2nd 6064 | . . . . . . . 8 | |
58 | ffvelrn 5553 | . . . . . . . 8 | |
59 | 56, 57, 58 | syl2an 287 | . . . . . . 7 |
60 | 51, 59 | opelxpd 4572 | . . . . . 6 |
61 | 45 | adantr 274 | . . . . . . . . . 10 |
62 | 34 | ad2antrl 481 | . . . . . . . . . 10 |
63 | 49 | ad2antll 482 | . . . . . . . . . 10 |
64 | f1ocnvfvb 5681 | . . . . . . . . . 10 | |
65 | 61, 62, 63, 64 | syl3anc 1216 | . . . . . . . . 9 |
66 | eqcom 2141 | . . . . . . . . 9 | |
67 | eqcom 2141 | . . . . . . . . 9 | |
68 | 65, 66, 67 | 3bitr4g 222 | . . . . . . . 8 |
69 | 53 | adantr 274 | . . . . . . . . . 10 |
70 | 40 | ad2antrl 481 | . . . . . . . . . 10 |
71 | 57 | ad2antll 482 | . . . . . . . . . 10 |
72 | f1ocnvfvb 5681 | . . . . . . . . . 10 | |
73 | 69, 70, 71, 72 | syl3anc 1216 | . . . . . . . . 9 |
74 | eqcom 2141 | . . . . . . . . 9 | |
75 | eqcom 2141 | . . . . . . . . 9 | |
76 | 73, 74, 75 | 3bitr4g 222 | . . . . . . . 8 |
77 | 68, 76 | anbi12d 464 | . . . . . . 7 |
78 | eqop 6075 | . . . . . . . 8 | |
79 | 78 | ad2antll 482 | . . . . . . 7 |
80 | eqop 6075 | . . . . . . . 8 | |
81 | 80 | ad2antrl 481 | . . . . . . 7 |
82 | 77, 79, 81 | 3bitr4rd 220 | . . . . . 6 |
83 | 30, 43, 60, 82 | f1ocnv2d 5974 | . . . . 5 |
84 | 83 | simprd 113 | . . . 4 |
85 | vex 2689 | . . . . . . . 8 | |
86 | vex 2689 | . . . . . . . 8 | |
87 | 85, 86 | op1std 6046 | . . . . . . 7 |
88 | 87 | fveq2d 5425 | . . . . . 6 |
89 | 85, 86 | op2ndd 6047 | . . . . . . 7 |
90 | 89 | fveq2d 5425 | . . . . . 6 |
91 | 88, 90 | opeq12d 3713 | . . . . 5 |
92 | 91 | mpompt 5863 | . . . 4 |
93 | 84, 92 | syl6eq 2188 | . . 3 |
94 | cntop2 12371 | . . . . . 6 | |
95 | 3, 94 | syl 14 | . . . . 5 |
96 | 31 | toptopon 12185 | . . . . 5 TopOn |
97 | 95, 96 | sylib 121 | . . . 4 TopOn |
98 | cntop2 12371 | . . . . . 6 | |
99 | 11, 98 | syl 14 | . . . . 5 |
100 | 37 | toptopon 12185 | . . . . 5 TopOn |
101 | 99, 100 | sylib 121 | . . . 4 TopOn |
102 | 97, 101 | cnmpt1st 12457 | . . . . 5 |
103 | hmeocnvcn 12475 | . . . . . 6 | |
104 | 1, 103 | syl 14 | . . . . 5 |
105 | 97, 101, 102, 104 | cnmpt21f 12461 | . . . 4 |
106 | 97, 101 | cnmpt2nd 12458 | . . . . 5 |
107 | hmeocnvcn 12475 | . . . . . 6 | |
108 | 9, 107 | syl 14 | . . . . 5 |
109 | 97, 101, 106, 108 | cnmpt21f 12461 | . . . 4 |
110 | 97, 101, 105, 109 | cnmpt2t 12462 | . . 3 |
111 | 93, 110 | eqeltrd 2216 | . 2 |
112 | ishmeo 12473 | . 2 | |
113 | 21, 111, 112 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cop 3530 cuni 3736 cmpt 3989 cxp 4537 ccnv 4538 wf 5119 wf1o 5122 cfv 5123 (class class class)co 5774 cmpo 5776 c1st 6036 c2nd 6037 ctop 12164 TopOnctopon 12177 ccn 12354 ctx 12421 chmeo 12469 |
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 df-hmeo 12470 |
This theorem is referenced by: (None) |
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