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Mirrors > Home > MPE Home > Th. List > pntlemc | Structured version Unicode version |
Description: Lemma for pnt 22989.
Closure for the constants used in the proof. For
comparison with Equation 10.6.27 of [Shapiro], p. 434, ![]() ![]() ![]() |
Ref | Expression |
---|---|
pntlem1.r |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pntlem1.a |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pntlem1.b |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pntlem1.l |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pntlem1.d |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pntlem1.f |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pntlem1.u |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pntlem1.u2 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pntlem1.e |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pntlem1.k |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
pntlemc |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pntlem1.e |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | pntlem1.u |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | pntlem1.r |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | pntlem1.a |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | pntlem1.b |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | pntlem1.l |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | pntlem1.d |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | pntlem1.f |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 3, 4, 5, 6, 7, 8 | pntlemd 22969 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 9 | simp2d 1001 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 2, 10 | rpdivcld 11148 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 1, 11 | syl5eqel 2543 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | pntlem1.k |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 5, 12 | rpdivcld 11148 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 14 | rpred 11131 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15 | rpefcld 13500 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 13, 16 | syl5eqel 2543 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 12 | rpred 11131 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 12 | rpgt0d 11134 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 2 | rpred 11131 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 4 | rpred 11131 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 10 | rpred 11131 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | pntlem1.u2 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 21 | ltp1d 10367 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24, 7 | syl6breqr 4433 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 20, 21, 22, 23, 25 | lelttrd 9633 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 10 | rpcnd 11133 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 27 | mulid1d 9507 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 26, 28 | breqtrrd 4419 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 1red 9505 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 20, 30, 10 | ltdivmuld 11178 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 29, 31 | mpbird 232 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 1, 32 | syl5eqbr 4426 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 0xr 9534 |
. . . . 5
![]() ![]() ![]() ![]() | |
35 | 1re 9489 |
. . . . . 6
![]() ![]() ![]() ![]() | |
36 | 35 | rexri 9540 |
. . . . 5
![]() ![]() ![]() ![]() |
37 | elioo2 11445 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
38 | 34, 36, 37 | mp2an 672 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 18, 19, 33, 38 | syl3anbrc 1172 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | efgt1 13511 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
41 | 14, 40 | syl 16 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 41, 13 | syl6breqr 4433 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | ltaddrp 11127 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
44 | 35, 4, 43 | sylancr 663 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 2 | rpcnne0d 11140 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | divid 10125 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
47 | 45, 46 | syl 16 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 4 | rpcnd 11133 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | ax-1cn 9444 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
50 | addcom 9659 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
51 | 48, 49, 50 | sylancl 662 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 7, 51 | syl5eq 2504 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 44, 47, 52 | 3brtr4d 4423 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | 20, 2, 10, 53 | ltdiv23d 11187 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 1, 54 | syl5eqbr 4426 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 | difrp 11128 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
57 | 18, 20, 56 | syl2anc 661 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
58 | 55, 57 | mpbid 210 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
59 | 39, 42, 58 | 3jca 1168 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | 12, 17, 59 | 3jca 1168 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-inf2 7951 ax-cnex 9442 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-mulcom 9450 ax-addass 9451 ax-mulass 9452 ax-distr 9453 ax-i2m1 9454 ax-1ne0 9455 ax-1rid 9456 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 ax-pre-lttri 9460 ax-pre-lttrn 9461 ax-pre-ltadd 9462 ax-pre-mulgt0 9463 ax-pre-sup 9464 ax-addf 9465 ax-mulf 9466 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-tp 3983 df-op 3985 df-uni 4193 df-int 4230 df-iun 4274 df-br 4394 df-opab 4452 df-mpt 4453 df-tr 4487 df-eprel 4733 df-id 4737 df-po 4742 df-so 4743 df-fr 4780 df-se 4781 df-we 4782 df-ord 4823 df-on 4824 df-lim 4825 df-suc 4826 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-isom 5528 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-om 6580 df-1st 6680 df-2nd 6681 df-recs 6935 df-rdg 6969 df-1o 7023 df-oadd 7027 df-er 7204 df-pm 7320 df-en 7414 df-dom 7415 df-sdom 7416 df-fin 7417 df-sup 7795 df-oi 7828 df-card 8213 df-pnf 9524 df-mnf 9525 df-xr 9526 df-ltxr 9527 df-le 9528 df-sub 9701 df-neg 9702 df-div 10098 df-nn 10427 df-2 10484 df-3 10485 df-4 10486 df-5 10487 df-6 10488 df-7 10489 df-8 10490 df-9 10491 df-10 10492 df-n0 10684 df-z 10751 df-dec 10860 df-uz 10966 df-rp 11096 df-ioo 11408 df-ico 11410 df-fz 11548 df-fzo 11659 df-fl 11752 df-seq 11917 df-exp 11976 df-fac 12162 df-bc 12189 df-hash 12214 df-shft 12667 df-cj 12699 df-re 12700 df-im 12701 df-sqr 12835 df-abs 12836 df-limsup 13060 df-clim 13077 df-rlim 13078 df-sum 13275 df-ef 13464 |
This theorem is referenced by: pntlema 22971 pntlemb 22972 pntlemg 22973 pntlemh 22974 pntlemq 22976 pntlemr 22977 pntlemj 22978 pntlemi 22979 pntlemf 22980 pntlemo 22982 pntleme 22983 pntlemp 22985 |
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