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Mirrors > Home > MPE Home > Th. List > Mathboxes > hspval | Structured version Visualization version Unicode version |
Description: The value of the half-space of n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
hspval.h | |
hspval.x | |
hspval.i | |
hspval.y |
Ref | Expression |
---|---|
hspval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hspval.h | . . . 4 | |
2 | 1 | a1i 11 | . . 3 |
3 | id 22 | . . . . 5 | |
4 | eqidd 2623 | . . . . 5 | |
5 | ixpeq1 7919 | . . . . 5 | |
6 | 3, 4, 5 | mpt2eq123dv 6717 | . . . 4 |
7 | 6 | adantl 482 | . . 3 |
8 | hspval.x | . . 3 | |
9 | reex 10027 | . . . . 5 | |
10 | 9 | a1i 11 | . . . 4 |
11 | eqid 2622 | . . . . 5 | |
12 | 11 | mpt2exg 7245 | . . . 4 |
13 | 8, 10, 12 | syl2anc 693 | . . 3 |
14 | 2, 7, 8, 13 | fvmptd 6288 | . 2 |
15 | simpl 473 | . . . . . 6 | |
16 | 15 | eqeq2d 2632 | . . . . 5 |
17 | simpr 477 | . . . . . 6 | |
18 | 17 | oveq2d 6666 | . . . . 5 |
19 | 16, 18 | ifbieq1d 4109 | . . . 4 |
20 | 19 | ixpeq2dv 7924 | . . 3 |
21 | 20 | adantl 482 | . 2 |
22 | hspval.i | . 2 | |
23 | hspval.y | . 2 | |
24 | ovex 6678 | . . . . . 6 | |
25 | 24, 9 | keepel 4155 | . . . . 5 |
26 | 25 | a1i 11 | . . . 4 |
27 | 26 | ralrimiva 2966 | . . 3 |
28 | ixpexg 7932 | . . 3 | |
29 | 27, 28 | syl 17 | . 2 |
30 | 14, 21, 22, 23, 29 | ovmpt2d 6788 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 cvv 3200 cif 4086 cmpt 4729 cfv 5888 (class class class)co 6650 cmpt2 6652 cixp 7908 cfn 7955 cr 9935 cmnf 10072 cioo 12175 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 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-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-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-ixp 7909 |
This theorem is referenced by: hspdifhsp 40830 hspmbllem2 40841 hspmbl 40843 |
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