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Mirrors > Home > ILE Home > Th. List > neitx | Unicode version |
Description: The Cartesian product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.) |
Ref | Expression |
---|---|
neitx.x | |
neitx.y |
Ref | Expression |
---|---|
neitx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neitx.x | . . . . . 6 | |
2 | 1 | neii1 12316 | . . . . 5 |
3 | 2 | ad2ant2r 500 | . . . 4 |
4 | neitx.y | . . . . . 6 | |
5 | 4 | neii1 12316 | . . . . 5 |
6 | 5 | ad2ant2l 499 | . . . 4 |
7 | xpss12 4646 | . . . 4 | |
8 | 3, 6, 7 | syl2anc 408 | . . 3 |
9 | 1, 4 | txuni 12432 | . . . 4 |
10 | 9 | adantr 274 | . . 3 |
11 | 8, 10 | sseqtrd 3135 | . 2 |
12 | simp-5l 532 | . . . . . 6 | |
13 | simp-4r 531 | . . . . . 6 | |
14 | simplr 519 | . . . . . 6 | |
15 | txopn 12434 | . . . . . 6 | |
16 | 12, 13, 14, 15 | syl12anc 1214 | . . . . 5 |
17 | simpr1l 1038 | . . . . . . 7 | |
18 | 17 | 3anassrs 1207 | . . . . . 6 |
19 | simprl 520 | . . . . . 6 | |
20 | xpss12 4646 | . . . . . 6 | |
21 | 18, 19, 20 | syl2anc 408 | . . . . 5 |
22 | simpr1r 1039 | . . . . . . 7 | |
23 | 22 | 3anassrs 1207 | . . . . . 6 |
24 | simprr 521 | . . . . . 6 | |
25 | xpss12 4646 | . . . . . 6 | |
26 | 23, 24, 25 | syl2anc 408 | . . . . 5 |
27 | sseq2 3121 | . . . . . . 7 | |
28 | sseq1 3120 | . . . . . . 7 | |
29 | 27, 28 | anbi12d 464 | . . . . . 6 |
30 | 29 | rspcev 2789 | . . . . 5 |
31 | 16, 21, 26, 30 | syl12anc 1214 | . . . 4 |
32 | neii2 12318 | . . . . . 6 | |
33 | 32 | ad2ant2l 499 | . . . . 5 |
34 | 33 | ad2antrr 479 | . . . 4 |
35 | 31, 34 | r19.29a 2575 | . . 3 |
36 | neii2 12318 | . . . 4 | |
37 | 36 | ad2ant2r 500 | . . 3 |
38 | 35, 37 | r19.29a 2575 | . 2 |
39 | txtop 12429 | . . . 4 | |
40 | 39 | adantr 274 | . . 3 |
41 | 1 | neiss2 12311 | . . . . . 6 |
42 | 41 | ad2ant2r 500 | . . . . 5 |
43 | 4 | neiss2 12311 | . . . . . 6 |
44 | 43 | ad2ant2l 499 | . . . . 5 |
45 | xpss12 4646 | . . . . 5 | |
46 | 42, 44, 45 | syl2anc 408 | . . . 4 |
47 | 46, 10 | sseqtrd 3135 | . . 3 |
48 | eqid 2139 | . . . 4 | |
49 | 48 | isnei 12313 | . . 3 |
50 | 40, 47, 49 | syl2anc 408 | . 2 |
51 | 11, 38, 50 | mpbir2and 928 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wrex 2417 wss 3071 cuni 3736 cxp 4537 cfv 5123 (class class class)co 5774 ctop 12164 cnei 12307 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-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-topgen 12141 df-top 12165 df-topon 12178 df-bases 12210 df-nei 12308 df-tx 12422 |
This theorem is referenced by: (None) |
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