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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem37 | Structured version Visualization version Unicode version |
Description: This lemma is used to
prove the existence of a function p as in Lemma 1
of [BrosowskiDeutsh] p. 90: p
is in the subalgebra, such that 0 <= p <=
1, p(t_0) = 0, and p > 0 on T - U.
Z is used for t0, P is used for p,
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Ref | Expression |
---|---|
stoweidlem37.1 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
stoweidlem37.2 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
stoweidlem37.3 |
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stoweidlem37.4 |
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stoweidlem37.5 |
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stoweidlem37.6 |
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Ref | Expression |
---|---|
stoweidlem37 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stoweidlem37.6 |
. . 3
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2 | stoweidlem37.1 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | stoweidlem37.2 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | stoweidlem37.3 |
. . . 4
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5 | stoweidlem37.4 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | stoweidlem37.5 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 2, 3, 4, 5, 6 | stoweidlem30 37885 |
. . 3
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8 | 1, 7 | mpdan 673 |
. 2
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9 | 5 | fnvinran 37329 |
. . . . . . 7
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10 | fveq1 5862 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | eqeq1d 2452 |
. . . . . . . . 9
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12 | fveq1 5862 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 12 | breq2d 4413 |
. . . . . . . . . . 11
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14 | 12 | breq1d 4411 |
. . . . . . . . . . 11
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15 | 13, 14 | anbi12d 716 |
. . . . . . . . . 10
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16 | 15 | ralbidv 2826 |
. . . . . . . . 9
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17 | 11, 16 | anbi12d 716 |
. . . . . . . 8
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18 | 17, 2 | elrab2 3197 |
. . . . . . 7
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19 | 9, 18 | sylib 200 |
. . . . . 6
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20 | 19 | simprld 764 |
. . . . 5
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21 | 20 | sumeq2dv 13762 |
. . . 4
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22 | fzfi 12182 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | olc 386 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | sumz 13781 |
. . . . 5
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25 | 22, 23, 24 | mp2b 10 |
. . . 4
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26 | 21, 25 | syl6eq 2500 |
. . 3
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27 | 26 | oveq2d 6304 |
. 2
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28 | 4 | nncnd 10622 |
. . . 4
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29 | 4 | nnne0d 10651 |
. . . 4
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30 | 28, 29 | reccld 10373 |
. . 3
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31 | 30 | mul01d 9829 |
. 2
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32 | 8, 27, 31 | 3eqtrd 2488 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-inf2 8143 ax-cnex 9592 ax-resscn 9593 ax-1cn 9594 ax-icn 9595 ax-addcl 9596 ax-addrcl 9597 ax-mulcl 9598 ax-mulrcl 9599 ax-mulcom 9600 ax-addass 9601 ax-mulass 9602 ax-distr 9603 ax-i2m1 9604 ax-1ne0 9605 ax-1rid 9606 ax-rnegex 9607 ax-rrecex 9608 ax-cnre 9609 ax-pre-lttri 9610 ax-pre-lttrn 9611 ax-pre-ltadd 9612 ax-pre-mulgt0 9613 ax-pre-sup 9614 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-fal 1449 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-int 4234 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-se 4793 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-isom 5590 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-om 6690 df-1st 6790 df-2nd 6791 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-1o 7179 df-oadd 7183 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-fin 7570 df-sup 7953 df-oi 8022 df-card 8370 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-sub 9859 df-neg 9860 df-div 10267 df-nn 10607 df-2 10665 df-3 10666 df-n0 10867 df-z 10935 df-uz 11157 df-rp 11300 df-fz 11782 df-fzo 11913 df-seq 12211 df-exp 12270 df-hash 12513 df-cj 13155 df-re 13156 df-im 13157 df-sqrt 13291 df-abs 13292 df-clim 13545 df-sum 13746 |
This theorem is referenced by: stoweidlem44 37899 |
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