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Mirrors > Home > ILE Home > Th. List > xrminmax | Unicode version |
Description: Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.) |
Ref | Expression |
---|---|
xrminmax | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnegcl 9615 | . . . . . . . . . . . 12 | |
2 | elprg 3547 | . . . . . . . . . . . 12 | |
3 | 1, 2 | syl 14 | . . . . . . . . . . 11 |
4 | 3 | adantl 275 | . . . . . . . . . 10 |
5 | simpr 109 | . . . . . . . . . . . . 13 | |
6 | simpll 518 | . . . . . . . . . . . . 13 | |
7 | 5, 6 | xrnegcon1d 11033 | . . . . . . . . . . . 12 |
8 | eqcom 2141 | . . . . . . . . . . . 12 | |
9 | 7, 8 | syl6bb 195 | . . . . . . . . . . 11 |
10 | simplr 519 | . . . . . . . . . . . . 13 | |
11 | 5, 10 | xrnegcon1d 11033 | . . . . . . . . . . . 12 |
12 | eqcom 2141 | . . . . . . . . . . . 12 | |
13 | 11, 12 | syl6bb 195 | . . . . . . . . . . 11 |
14 | 9, 13 | orbi12d 782 | . . . . . . . . . 10 |
15 | 4, 14 | bitrd 187 | . . . . . . . . 9 |
16 | 15 | rabbidva 2674 | . . . . . . . 8 |
17 | dfrab2 3351 | . . . . . . . . . 10 | |
18 | dfpr2 3546 | . . . . . . . . . . 11 | |
19 | 18 | ineq1i 3273 | . . . . . . . . . 10 |
20 | 17, 19 | eqtr4i 2163 | . . . . . . . . 9 |
21 | xnegcl 9615 | . . . . . . . . . . 11 | |
22 | xnegcl 9615 | . . . . . . . . . . 11 | |
23 | prssi 3678 | . . . . . . . . . . 11 | |
24 | 21, 22, 23 | syl2an 287 | . . . . . . . . . 10 |
25 | df-ss 3084 | . . . . . . . . . 10 | |
26 | 24, 25 | sylib 121 | . . . . . . . . 9 |
27 | 20, 26 | syl5eq 2184 | . . . . . . . 8 |
28 | 16, 27 | eqtrd 2172 | . . . . . . 7 |
29 | 28 | supeq1d 6874 | . . . . . 6 |
30 | xrmaxcl 11021 | . . . . . . 7 | |
31 | 21, 22, 30 | syl2an 287 | . . . . . 6 |
32 | 29, 31 | eqeltrd 2216 | . . . . 5 |
33 | 32 | xnegcld 9638 | . . . 4 |
34 | xnegeq 9610 | . . . . . . . . 9 | |
35 | 34 | adantl 275 | . . . . . . . 8 |
36 | xrmax1sup 11022 | . . . . . . . . . 10 | |
37 | 21, 22, 36 | syl2an 287 | . . . . . . . . 9 |
38 | 37 | ad2antrr 479 | . . . . . . . 8 |
39 | 35, 38 | eqbrtrd 3950 | . . . . . . 7 |
40 | simpll 518 | . . . . . . . 8 | |
41 | simpr 109 | . . . . . . . . 9 | |
42 | simplll 522 | . . . . . . . . 9 | |
43 | 41, 42 | eqeltrd 2216 | . . . . . . . 8 |
44 | xnegeq 9610 | . . . . . . . . . . . . 13 | |
45 | 29, 44 | syl 14 | . . . . . . . . . . . 12 |
46 | 45 | breq2d 3941 | . . . . . . . . . . 11 |
47 | 46 | notbid 656 | . . . . . . . . . 10 |
48 | 47 | adantr 274 | . . . . . . . . 9 |
49 | 31 | adantr 274 | . . . . . . . . . . 11 |
50 | 49 | xnegcld 9638 | . . . . . . . . . 10 |
51 | xrlenlt 7829 | . . . . . . . . . 10 | |
52 | 50, 51 | sylancom 416 | . . . . . . . . 9 |
53 | xleneg 9620 | . . . . . . . . . . 11 | |
54 | 50, 53 | sylancom 416 | . . . . . . . . . 10 |
55 | xnegneg 9616 | . . . . . . . . . . . 12 | |
56 | 49, 55 | syl 14 | . . . . . . . . . . 11 |
57 | 56 | breq2d 3941 | . . . . . . . . . 10 |
58 | 54, 57 | bitrd 187 | . . . . . . . . 9 |
59 | 48, 52, 58 | 3bitr2d 215 | . . . . . . . 8 |
60 | 40, 43, 59 | syl2anc 408 | . . . . . . 7 |
61 | 39, 60 | mpbird 166 | . . . . . 6 |
62 | xnegeq 9610 | . . . . . . . . 9 | |
63 | 62 | adantl 275 | . . . . . . . 8 |
64 | xrmax2sup 11023 | . . . . . . . . . 10 | |
65 | 21, 22, 64 | syl2an 287 | . . . . . . . . 9 |
66 | 65 | ad2antrr 479 | . . . . . . . 8 |
67 | 63, 66 | eqbrtrd 3950 | . . . . . . 7 |
68 | simpll 518 | . . . . . . . 8 | |
69 | simpr 109 | . . . . . . . . 9 | |
70 | simpllr 523 | . . . . . . . . 9 | |
71 | 69, 70 | eqeltrd 2216 | . . . . . . . 8 |
72 | 68, 71, 59 | syl2anc 408 | . . . . . . 7 |
73 | 67, 72 | mpbird 166 | . . . . . 6 |
74 | elpri 3550 | . . . . . . 7 | |
75 | 74 | adantl 275 | . . . . . 6 |
76 | 61, 73, 75 | mpjaodan 787 | . . . . 5 |
77 | 76 | ralrimiva 2505 | . . . 4 |
78 | 21 | ad3antrrr 483 | . . . . . . . . 9 |
79 | 22 | ad3antlr 484 | . . . . . . . . 9 |
80 | simplr 519 | . . . . . . . . . 10 | |
81 | 80 | xnegcld 9638 | . . . . . . . . 9 |
82 | simpr 109 | . . . . . . . . . . 11 | |
83 | 45 | breq1d 3939 | . . . . . . . . . . . 12 |
84 | 83 | ad2antrr 479 | . . . . . . . . . . 11 |
85 | 82, 84 | mpbid 146 | . . . . . . . . . 10 |
86 | 50 | adantr 274 | . . . . . . . . . . . 12 |
87 | xltneg 9619 | . . . . . . . . . . . 12 | |
88 | 86, 80, 87 | syl2anc 408 | . . . . . . . . . . 11 |
89 | 56 | breq2d 3941 | . . . . . . . . . . . 12 |
90 | 89 | adantr 274 | . . . . . . . . . . 11 |
91 | 88, 90 | bitrd 187 | . . . . . . . . . 10 |
92 | 85, 91 | mpbid 146 | . . . . . . . . 9 |
93 | xrmaxleastlt 11025 | . . . . . . . . 9 | |
94 | 78, 79, 81, 92, 93 | syl22anc 1217 | . . . . . . . 8 |
95 | simplll 522 | . . . . . . . . . 10 | |
96 | xltneg 9619 | . . . . . . . . . 10 | |
97 | 95, 80, 96 | syl2anc 408 | . . . . . . . . 9 |
98 | simpllr 523 | . . . . . . . . . 10 | |
99 | xltneg 9619 | . . . . . . . . . 10 | |
100 | 98, 80, 99 | syl2anc 408 | . . . . . . . . 9 |
101 | 97, 100 | orbi12d 782 | . . . . . . . 8 |
102 | 94, 101 | mpbird 166 | . . . . . . 7 |
103 | breq1 3932 | . . . . . . . . 9 | |
104 | breq1 3932 | . . . . . . . . 9 | |
105 | 103, 104 | rexprg 3575 | . . . . . . . 8 |
106 | 105 | ad2antrr 479 | . . . . . . 7 |
107 | 102, 106 | mpbird 166 | . . . . . 6 |
108 | 107 | ex 114 | . . . . 5 |
109 | 108 | ralrimiva 2505 | . . . 4 |
110 | breq2 3933 | . . . . . . . 8 | |
111 | 110 | notbid 656 | . . . . . . 7 |
112 | 111 | ralbidv 2437 | . . . . . 6 |
113 | breq1 3932 | . . . . . . . 8 | |
114 | 113 | imbi1d 230 | . . . . . . 7 |
115 | 114 | ralbidv 2437 | . . . . . 6 |
116 | 112, 115 | anbi12d 464 | . . . . 5 |
117 | 116 | rspcev 2789 | . . . 4 |
118 | 33, 77, 109, 117 | syl12anc 1214 | . . 3 |
119 | prssi 3678 | . . 3 | |
120 | 118, 119 | infxrnegsupex 11032 | . 2 inf |
121 | 120, 45 | eqtrd 2172 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 cab 2125 wral 2416 wrex 2417 crab 2420 cin 3070 wss 3071 cpr 3528 class class class wbr 3929 csup 6869 infcinf 6870 cxr 7799 clt 7800 cle 7801 cxne 9556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-xneg 9559 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 |
This theorem is referenced by: xrmincl 11035 xrmin1inf 11036 xrmin2inf 11037 xrmineqinf 11038 xrltmininf 11039 xrlemininf 11040 xrminltinf 11041 xrminrecl 11042 xrminrpcl 11043 xrminadd 11044 |
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