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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 9759 | . . . . . . . . . . . 12 | |
2 | elprg 3590 | . . . . . . . . . . . 12 | |
3 | 1, 2 | syl 14 | . . . . . . . . . . 11 |
4 | 3 | adantl 275 | . . . . . . . . . 10 |
5 | simpr 109 | . . . . . . . . . . . . 13 | |
6 | simpll 519 | . . . . . . . . . . . . 13 | |
7 | 5, 6 | xrnegcon1d 11191 | . . . . . . . . . . . 12 |
8 | eqcom 2166 | . . . . . . . . . . . 12 | |
9 | 7, 8 | bitrdi 195 | . . . . . . . . . . 11 |
10 | simplr 520 | . . . . . . . . . . . . 13 | |
11 | 5, 10 | xrnegcon1d 11191 | . . . . . . . . . . . 12 |
12 | eqcom 2166 | . . . . . . . . . . . 12 | |
13 | 11, 12 | bitrdi 195 | . . . . . . . . . . 11 |
14 | 9, 13 | orbi12d 783 | . . . . . . . . . 10 |
15 | 4, 14 | bitrd 187 | . . . . . . . . 9 |
16 | 15 | rabbidva 2709 | . . . . . . . 8 |
17 | dfrab2 3392 | . . . . . . . . . 10 | |
18 | dfpr2 3589 | . . . . . . . . . . 11 | |
19 | 18 | ineq1i 3314 | . . . . . . . . . 10 |
20 | 17, 19 | eqtr4i 2188 | . . . . . . . . 9 |
21 | xnegcl 9759 | . . . . . . . . . . 11 | |
22 | xnegcl 9759 | . . . . . . . . . . 11 | |
23 | prssi 3725 | . . . . . . . . . . 11 | |
24 | 21, 22, 23 | syl2an 287 | . . . . . . . . . 10 |
25 | df-ss 3124 | . . . . . . . . . 10 | |
26 | 24, 25 | sylib 121 | . . . . . . . . 9 |
27 | 20, 26 | syl5eq 2209 | . . . . . . . 8 |
28 | 16, 27 | eqtrd 2197 | . . . . . . 7 |
29 | 28 | supeq1d 6943 | . . . . . 6 |
30 | xrmaxcl 11179 | . . . . . . 7 | |
31 | 21, 22, 30 | syl2an 287 | . . . . . 6 |
32 | 29, 31 | eqeltrd 2241 | . . . . 5 |
33 | 32 | xnegcld 9782 | . . . 4 |
34 | xnegeq 9754 | . . . . . . . . 9 | |
35 | 34 | adantl 275 | . . . . . . . 8 |
36 | xrmax1sup 11180 | . . . . . . . . . 10 | |
37 | 21, 22, 36 | syl2an 287 | . . . . . . . . 9 |
38 | 37 | ad2antrr 480 | . . . . . . . 8 |
39 | 35, 38 | eqbrtrd 3998 | . . . . . . 7 |
40 | simpll 519 | . . . . . . . 8 | |
41 | simpr 109 | . . . . . . . . 9 | |
42 | simplll 523 | . . . . . . . . 9 | |
43 | 41, 42 | eqeltrd 2241 | . . . . . . . 8 |
44 | xnegeq 9754 | . . . . . . . . . . . . 13 | |
45 | 29, 44 | syl 14 | . . . . . . . . . . . 12 |
46 | 45 | breq2d 3988 | . . . . . . . . . . 11 |
47 | 46 | notbid 657 | . . . . . . . . . 10 |
48 | 47 | adantr 274 | . . . . . . . . 9 |
49 | 31 | adantr 274 | . . . . . . . . . . 11 |
50 | 49 | xnegcld 9782 | . . . . . . . . . 10 |
51 | xrlenlt 7954 | . . . . . . . . . 10 | |
52 | 50, 51 | sylancom 417 | . . . . . . . . 9 |
53 | xleneg 9764 | . . . . . . . . . . 11 | |
54 | 50, 53 | sylancom 417 | . . . . . . . . . 10 |
55 | xnegneg 9760 | . . . . . . . . . . . 12 | |
56 | 49, 55 | syl 14 | . . . . . . . . . . 11 |
57 | 56 | breq2d 3988 | . . . . . . . . . 10 |
58 | 54, 57 | bitrd 187 | . . . . . . . . 9 |
59 | 48, 52, 58 | 3bitr2d 215 | . . . . . . . 8 |
60 | 40, 43, 59 | syl2anc 409 | . . . . . . 7 |
61 | 39, 60 | mpbird 166 | . . . . . 6 |
62 | xnegeq 9754 | . . . . . . . . 9 | |
63 | 62 | adantl 275 | . . . . . . . 8 |
64 | xrmax2sup 11181 | . . . . . . . . . 10 | |
65 | 21, 22, 64 | syl2an 287 | . . . . . . . . 9 |
66 | 65 | ad2antrr 480 | . . . . . . . 8 |
67 | 63, 66 | eqbrtrd 3998 | . . . . . . 7 |
68 | simpll 519 | . . . . . . . 8 | |
69 | simpr 109 | . . . . . . . . 9 | |
70 | simpllr 524 | . . . . . . . . 9 | |
71 | 69, 70 | eqeltrd 2241 | . . . . . . . 8 |
72 | 68, 71, 59 | syl2anc 409 | . . . . . . 7 |
73 | 67, 72 | mpbird 166 | . . . . . 6 |
74 | elpri 3593 | . . . . . . 7 | |
75 | 74 | adantl 275 | . . . . . 6 |
76 | 61, 73, 75 | mpjaodan 788 | . . . . 5 |
77 | 76 | ralrimiva 2537 | . . . 4 |
78 | 21 | ad3antrrr 484 | . . . . . . . . 9 |
79 | 22 | ad3antlr 485 | . . . . . . . . 9 |
80 | simplr 520 | . . . . . . . . . 10 | |
81 | 80 | xnegcld 9782 | . . . . . . . . 9 |
82 | simpr 109 | . . . . . . . . . . 11 | |
83 | 45 | breq1d 3986 | . . . . . . . . . . . 12 |
84 | 83 | ad2antrr 480 | . . . . . . . . . . 11 |
85 | 82, 84 | mpbid 146 | . . . . . . . . . 10 |
86 | 50 | adantr 274 | . . . . . . . . . . . 12 |
87 | xltneg 9763 | . . . . . . . . . . . 12 | |
88 | 86, 80, 87 | syl2anc 409 | . . . . . . . . . . 11 |
89 | 56 | breq2d 3988 | . . . . . . . . . . . 12 |
90 | 89 | adantr 274 | . . . . . . . . . . 11 |
91 | 88, 90 | bitrd 187 | . . . . . . . . . 10 |
92 | 85, 91 | mpbid 146 | . . . . . . . . 9 |
93 | xrmaxleastlt 11183 | . . . . . . . . 9 | |
94 | 78, 79, 81, 92, 93 | syl22anc 1228 | . . . . . . . 8 |
95 | simplll 523 | . . . . . . . . . 10 | |
96 | xltneg 9763 | . . . . . . . . . 10 | |
97 | 95, 80, 96 | syl2anc 409 | . . . . . . . . 9 |
98 | simpllr 524 | . . . . . . . . . 10 | |
99 | xltneg 9763 | . . . . . . . . . 10 | |
100 | 98, 80, 99 | syl2anc 409 | . . . . . . . . 9 |
101 | 97, 100 | orbi12d 783 | . . . . . . . 8 |
102 | 94, 101 | mpbird 166 | . . . . . . 7 |
103 | breq1 3979 | . . . . . . . . 9 | |
104 | breq1 3979 | . . . . . . . . 9 | |
105 | 103, 104 | rexprg 3622 | . . . . . . . 8 |
106 | 105 | ad2antrr 480 | . . . . . . 7 |
107 | 102, 106 | mpbird 166 | . . . . . 6 |
108 | 107 | ex 114 | . . . . 5 |
109 | 108 | ralrimiva 2537 | . . . 4 |
110 | breq2 3980 | . . . . . . . 8 | |
111 | 110 | notbid 657 | . . . . . . 7 |
112 | 111 | ralbidv 2464 | . . . . . 6 |
113 | breq1 3979 | . . . . . . . 8 | |
114 | 113 | imbi1d 230 | . . . . . . 7 |
115 | 114 | ralbidv 2464 | . . . . . 6 |
116 | 112, 115 | anbi12d 465 | . . . . 5 |
117 | 116 | rspcev 2825 | . . . 4 |
118 | 33, 77, 109, 117 | syl12anc 1225 | . . 3 |
119 | prssi 3725 | . . 3 | |
120 | 118, 119 | infxrnegsupex 11190 | . 2 inf |
121 | 120, 45 | eqtrd 2197 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 cab 2150 wral 2442 wrex 2443 crab 2446 cin 3110 wss 3111 cpr 3571 class class class wbr 3976 csup 6938 infcinf 6939 cxr 7923 clt 7924 cle 7925 cxne 9696 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-rp 9581 df-xneg 9699 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 |
This theorem is referenced by: xrmincl 11193 xrmin1inf 11194 xrmin2inf 11195 xrmineqinf 11196 xrltmininf 11197 xrlemininf 11198 xrminltinf 11199 xrminrecl 11200 xrminrpcl 11201 xrminadd 11202 |
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