Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordtprsval | Structured version Visualization version Unicode version |
Description: Value of the order topology for a preset. (Contributed by Thierry Arnoux, 11-Sep-2015.) |
Ref | Expression |
---|---|
ordtNEW.b | |
ordtNEW.l | |
ordtposval.e | |
ordtposval.f |
Ref | Expression |
---|---|
ordtprsval | ordTop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtNEW.l | . . . 4 | |
2 | fvex 6201 | . . . . 5 | |
3 | 2 | inex1 4799 | . . . 4 |
4 | 1, 3 | eqeltri 2697 | . . 3 |
5 | eqid 2622 | . . . 4 | |
6 | eqid 2622 | . . . 4 | |
7 | eqid 2622 | . . . 4 | |
8 | 5, 6, 7 | ordtval 20993 | . . 3 ordTop |
9 | 4, 8 | ax-mp 5 | . 2 ordTop |
10 | ordtNEW.b | . . . . . . 7 | |
11 | 10, 1 | prsdm 29960 | . . . . . 6 |
12 | 11 | sneqd 4189 | . . . . 5 |
13 | rabeq 3192 | . . . . . . . . . 10 | |
14 | 11, 13 | syl 17 | . . . . . . . . 9 |
15 | 11, 14 | mpteq12dv 4733 | . . . . . . . 8 |
16 | 15 | rneqd 5353 | . . . . . . 7 |
17 | ordtposval.e | . . . . . . 7 | |
18 | 16, 17 | syl6eqr 2674 | . . . . . 6 |
19 | rabeq 3192 | . . . . . . . . . 10 | |
20 | 11, 19 | syl 17 | . . . . . . . . 9 |
21 | 11, 20 | mpteq12dv 4733 | . . . . . . . 8 |
22 | 21 | rneqd 5353 | . . . . . . 7 |
23 | ordtposval.f | . . . . . . 7 | |
24 | 22, 23 | syl6eqr 2674 | . . . . . 6 |
25 | 18, 24 | uneq12d 3768 | . . . . 5 |
26 | 12, 25 | uneq12d 3768 | . . . 4 |
27 | 26 | fveq2d 6195 | . . 3 |
28 | 27 | fveq2d 6195 | . 2 |
29 | 9, 28 | syl5eq 2668 | 1 ordTop |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wceq 1483 wcel 1990 crab 2916 cvv 3200 cun 3572 cin 3573 csn 4177 class class class wbr 4653 cmpt 4729 cxp 5112 cdm 5114 crn 5115 cfv 5888 cfi 8316 cbs 15857 cple 15948 ctg 16098 ordTopcordt 16159 cpreset 16926 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 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-iota 5851 df-fun 5890 df-fv 5896 df-ordt 16161 df-preset 16928 |
This theorem is referenced by: ordtcnvNEW 29966 ordtrest2NEW 29969 ordtconnlem1 29970 |
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