![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pntlemc | Structured version Visualization version Unicode version |
Description: Lemma for pnt 24500.
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 24480 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 9 | simp2d 1027 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 2, 10 | rpdivcld 11386 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 1, 11 | syl5eqel 2543 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | pntlem1.k |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 5, 12 | rpdivcld 11386 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 14 | rpred 11369 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15 | rpefcld 14207 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 13, 16 | syl5eqel 2543 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 12 | rpred 11369 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 12 | rpgt0d 11372 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 2 | rpred 11369 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 4 | rpred 11369 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 10 | rpred 11369 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | pntlem1.u2 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 21 | ltp1d 10564 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24, 7 | syl6breqr 4456 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 20, 21, 22, 23, 25 | lelttrd 9818 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 10 | rpcnd 11371 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 27 | mulid1d 9685 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 26, 28 | breqtrrd 4442 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 1red 9683 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 20, 30, 10 | ltdivmuld 11417 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 29, 31 | mpbird 240 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 1, 32 | syl5eqbr 4449 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 0xr 9712 |
. . . . 5
![]() ![]() ![]() ![]() | |
35 | 1re 9667 |
. . . . . 6
![]() ![]() ![]() ![]() | |
36 | 35 | rexri 9718 |
. . . . 5
![]() ![]() ![]() ![]() |
37 | elioo2 11705 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
38 | 34, 36, 37 | mp2an 683 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 18, 19, 33, 38 | syl3anbrc 1198 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | efgt1 14218 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
41 | 14, 40 | syl 17 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 41, 13 | syl6breqr 4456 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | ltaddrp 11364 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
44 | 35, 4, 43 | sylancr 674 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 2 | rpcnne0d 11378 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | divid 10324 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
47 | 45, 46 | syl 17 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 4 | rpcnd 11371 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | ax-1cn 9622 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
50 | addcom 9844 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
51 | 48, 49, 50 | sylancl 673 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 7, 51 | syl5eq 2507 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 44, 47, 52 | 3brtr4d 4446 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | 20, 2, 10, 53 | ltdiv23d 11431 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 1, 54 | syl5eqbr 4449 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 | difrp 11365 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
57 | 18, 20, 56 | syl2anc 671 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
58 | 55, 57 | mpbid 215 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
59 | 39, 42, 58 | 3jca 1194 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | 12, 17, 59 | 3jca 1194 |
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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-rep 4528 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 ax-inf2 8171 ax-cnex 9620 ax-resscn 9621 ax-1cn 9622 ax-icn 9623 ax-addcl 9624 ax-addrcl 9625 ax-mulcl 9626 ax-mulrcl 9627 ax-mulcom 9628 ax-addass 9629 ax-mulass 9630 ax-distr 9631 ax-i2m1 9632 ax-1ne0 9633 ax-1rid 9634 ax-rnegex 9635 ax-rrecex 9636 ax-cnre 9637 ax-pre-lttri 9638 ax-pre-lttrn 9639 ax-pre-ltadd 9640 ax-pre-mulgt0 9641 ax-pre-sup 9642 ax-addf 9643 ax-mulf 9644 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-fal 1460 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-nel 2635 df-ral 2753 df-rex 2754 df-reu 2755 df-rmo 2756 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-int 4248 df-iun 4293 df-br 4416 df-opab 4475 df-mpt 4476 df-tr 4511 df-eprel 4763 df-id 4767 df-po 4773 df-so 4774 df-fr 4811 df-se 4812 df-we 4813 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-pred 5398 df-ord 5444 df-on 5445 df-lim 5446 df-suc 5447 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-isom 5609 df-riota 6276 df-ov 6317 df-oprab 6318 df-mpt2 6319 df-om 6719 df-1st 6819 df-2nd 6820 df-wrecs 7053 df-recs 7115 df-rdg 7153 df-1o 7207 df-oadd 7211 df-er 7388 df-pm 7500 df-en 7595 df-dom 7596 df-sdom 7597 df-fin 7598 df-sup 7981 df-inf 7982 df-oi 8050 df-card 8398 df-pnf 9702 df-mnf 9703 df-xr 9704 df-ltxr 9705 df-le 9706 df-sub 9887 df-neg 9888 df-div 10297 df-nn 10637 df-2 10695 df-3 10696 df-4 10697 df-5 10698 df-6 10699 df-7 10700 df-8 10701 df-9 10702 df-10 10703 df-n0 10898 df-z 10966 df-dec 11080 df-uz 11188 df-rp 11331 df-ioo 11667 df-ico 11669 df-fz 11813 df-fzo 11946 df-fl 12059 df-seq 12245 df-exp 12304 df-fac 12491 df-bc 12519 df-hash 12547 df-shft 13178 df-cj 13210 df-re 13211 df-im 13212 df-sqrt 13346 df-abs 13347 df-limsup 13574 df-clim 13600 df-rlim 13601 df-sum 13801 df-ef 14169 |
This theorem is referenced by: pntlema 24482 pntlemb 24483 pntlemg 24484 pntlemh 24485 pntlemq 24487 pntlemr 24488 pntlemj 24489 pntlemi 24490 pntlemf 24491 pntlemo 24493 pntleme 24494 pntlemp 24496 |
Copyright terms: Public domain | W3C validator |