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Mirrors > Home > ILE Home > Th. List > xrmaxiflemab | Unicode version |
Description: Lemma for xrmaxif 11178. A variation of xrmaxleim 11171- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.) |
Ref | Expression |
---|---|
xrmaxiflemab.a | |
xrmaxiflemab.b | |
xrmaxiflemab.ab |
Ref | Expression |
---|---|
xrmaxiflemab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . 4 | |
2 | 1 | iftrued 3522 | . . 3 |
3 | 2, 1 | eqtr4d 2200 | . 2 |
4 | simpr 109 | . . . 4 | |
5 | 4 | iffalsed 3525 | . . 3 |
6 | xrmaxiflemab.ab | . . . . . . 7 | |
7 | 6 | ad2antrr 480 | . . . . . 6 |
8 | simpr 109 | . . . . . 6 | |
9 | 7, 8 | breqtrd 4002 | . . . . 5 |
10 | xrmaxiflemab.a | . . . . . . 7 | |
11 | nltmnf 9715 | . . . . . . 7 | |
12 | 10, 11 | syl 14 | . . . . . 6 |
13 | 12 | ad2antrr 480 | . . . . 5 |
14 | 9, 13 | pm2.21dd 610 | . . . 4 |
15 | simpr 109 | . . . . . 6 | |
16 | 15 | iffalsed 3525 | . . . . 5 |
17 | simpr 109 | . . . . . . . 8 | |
18 | 6 | ad3antrrr 484 | . . . . . . . 8 |
19 | 17, 18 | eqbrtrrd 4000 | . . . . . . 7 |
20 | xrmaxiflemab.b | . . . . . . . . 9 | |
21 | pnfnlt 9714 | . . . . . . . . 9 | |
22 | 20, 21 | syl 14 | . . . . . . . 8 |
23 | 22 | ad3antrrr 484 | . . . . . . 7 |
24 | 19, 23 | pm2.21dd 610 | . . . . . 6 |
25 | simpr 109 | . . . . . . . 8 | |
26 | 25 | iffalsed 3525 | . . . . . . 7 |
27 | simpr 109 | . . . . . . . . 9 | |
28 | 27 | iftrued 3522 | . . . . . . . 8 |
29 | simpr 109 | . . . . . . . . . 10 | |
30 | 29 | iffalsed 3525 | . . . . . . . . 9 |
31 | 25 | adantr 274 | . . . . . . . . . . . 12 |
32 | elxr 9703 | . . . . . . . . . . . . . 14 | |
33 | 10, 32 | sylib 121 | . . . . . . . . . . . . 13 |
34 | 33 | ad4antr 486 | . . . . . . . . . . . 12 |
35 | 31, 29, 34 | ecase23d 1339 | . . . . . . . . . . 11 |
36 | 4 | ad3antrrr 484 | . . . . . . . . . . . 12 |
37 | 15 | ad2antrr 480 | . . . . . . . . . . . 12 |
38 | elxr 9703 | . . . . . . . . . . . . . 14 | |
39 | 20, 38 | sylib 121 | . . . . . . . . . . . . 13 |
40 | 39 | ad4antr 486 | . . . . . . . . . . . 12 |
41 | 36, 37, 40 | ecase23d 1339 | . . . . . . . . . . 11 |
42 | 35, 41 | jca 304 | . . . . . . . . . 10 |
43 | 6 | ad4antr 486 | . . . . . . . . . . 11 |
44 | 35, 41, 43 | ltled 8008 | . . . . . . . . . 10 |
45 | maxleim 11133 | . . . . . . . . . 10 | |
46 | 42, 44, 45 | sylc 62 | . . . . . . . . 9 |
47 | 30, 46 | eqtrd 2197 | . . . . . . . 8 |
48 | xrmnfdc 9770 | . . . . . . . . . 10 DECID | |
49 | exmiddc 826 | . . . . . . . . . 10 DECID | |
50 | 10, 48, 49 | 3syl 17 | . . . . . . . . 9 |
51 | 50 | ad3antrrr 484 | . . . . . . . 8 |
52 | 28, 47, 51 | mpjaodan 788 | . . . . . . 7 |
53 | 26, 52 | eqtrd 2197 | . . . . . 6 |
54 | xrpnfdc 9769 | . . . . . . . 8 DECID | |
55 | exmiddc 826 | . . . . . . . 8 DECID | |
56 | 10, 54, 55 | 3syl 17 | . . . . . . 7 |
57 | 56 | ad2antrr 480 | . . . . . 6 |
58 | 24, 53, 57 | mpjaodan 788 | . . . . 5 |
59 | 16, 58 | eqtrd 2197 | . . . 4 |
60 | xrmnfdc 9770 | . . . . . 6 DECID | |
61 | exmiddc 826 | . . . . . 6 DECID | |
62 | 20, 60, 61 | 3syl 17 | . . . . 5 |
63 | 62 | adantr 274 | . . . 4 |
64 | 14, 59, 63 | mpjaodan 788 | . . 3 |
65 | 5, 64 | eqtrd 2197 | . 2 |
66 | xrpnfdc 9769 | . . 3 DECID | |
67 | exmiddc 826 | . . 3 DECID | |
68 | 20, 66, 67 | 3syl 17 | . 2 |
69 | 3, 65, 68 | mpjaodan 788 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 824 w3o 966 wceq 1342 wcel 2135 cif 3515 cpr 3571 class class class wbr 3976 csup 6938 cr 7743 cpnf 7921 cmnf 7922 cxr 7923 clt 7924 cle 7925 |
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-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-apti 7859 |
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-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-iota 5147 df-riota 5792 df-sup 6940 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 |
This theorem is referenced by: xrmaxiflemlub 11175 |
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