Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > elsx | Structured version Visualization version Unicode version |
Description: The cartesian product of two open sets is an element of the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
Ref | Expression |
---|---|
elsx | ×s |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . 6 | |
2 | 1 | txbasex 21369 | . . . . 5 |
3 | sssigagen 30208 | . . . . 5 sigaGen | |
4 | 2, 3 | syl 17 | . . . 4 sigaGen |
5 | 4 | adantr 481 | . . 3 sigaGen |
6 | eqid 2622 | . . . . . 6 | |
7 | xpeq1 5128 | . . . . . . . 8 | |
8 | 7 | eqeq2d 2632 | . . . . . . 7 |
9 | xpeq2 5129 | . . . . . . . 8 | |
10 | 9 | eqeq2d 2632 | . . . . . . 7 |
11 | 8, 10 | rspc2ev 3324 | . . . . . 6 |
12 | 6, 11 | mp3an3 1413 | . . . . 5 |
13 | xpexg 6960 | . . . . . 6 | |
14 | eqid 2622 | . . . . . . 7 | |
15 | 14 | elrnmpt2g 6772 | . . . . . 6 |
16 | 13, 15 | syl 17 | . . . . 5 |
17 | 12, 16 | mpbird 247 | . . . 4 |
18 | 17 | adantl 482 | . . 3 |
19 | 5, 18 | sseldd 3604 | . 2 sigaGen |
20 | 1 | sxval 30253 | . . 3 ×s sigaGen |
21 | 20 | adantr 481 | . 2 ×s sigaGen |
22 | 19, 21 | eleqtrrd 2704 | 1 ×s |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wrex 2913 cvv 3200 wss 3574 cxp 5112 crn 5115 cfv 5888 (class class class)co 6650 cmpt2 6652 sigaGencsigagen 30201 ×s csx 30251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-siga 30171 df-sigagen 30202 df-sx 30252 |
This theorem is referenced by: 1stmbfm 30322 2ndmbfm 30323 |
Copyright terms: Public domain | W3C validator |