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Mirrors > Home > ILE Home > Th. List > xrmaxiflemab | Unicode version |
Description: Lemma for xrmaxif 11020. A variation of xrmaxleim 11013- 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 3481 | . . 3 |
3 | 2, 1 | eqtr4d 2175 | . 2 |
4 | simpr 109 | . . . 4 | |
5 | 4 | iffalsed 3484 | . . 3 |
6 | xrmaxiflemab.ab | . . . . . . 7 | |
7 | 6 | ad2antrr 479 | . . . . . 6 |
8 | simpr 109 | . . . . . 6 | |
9 | 7, 8 | breqtrd 3954 | . . . . 5 |
10 | xrmaxiflemab.a | . . . . . . 7 | |
11 | nltmnf 9574 | . . . . . . 7 | |
12 | 10, 11 | syl 14 | . . . . . 6 |
13 | 12 | ad2antrr 479 | . . . . 5 |
14 | 9, 13 | pm2.21dd 609 | . . . 4 |
15 | simpr 109 | . . . . . 6 | |
16 | 15 | iffalsed 3484 | . . . . 5 |
17 | simpr 109 | . . . . . . . 8 | |
18 | 6 | ad3antrrr 483 | . . . . . . . 8 |
19 | 17, 18 | eqbrtrrd 3952 | . . . . . . 7 |
20 | xrmaxiflemab.b | . . . . . . . . 9 | |
21 | pnfnlt 9573 | . . . . . . . . 9 | |
22 | 20, 21 | syl 14 | . . . . . . . 8 |
23 | 22 | ad3antrrr 483 | . . . . . . 7 |
24 | 19, 23 | pm2.21dd 609 | . . . . . 6 |
25 | simpr 109 | . . . . . . . 8 | |
26 | 25 | iffalsed 3484 | . . . . . . 7 |
27 | simpr 109 | . . . . . . . . 9 | |
28 | 27 | iftrued 3481 | . . . . . . . 8 |
29 | simpr 109 | . . . . . . . . . 10 | |
30 | 29 | iffalsed 3484 | . . . . . . . . 9 |
31 | 25 | adantr 274 | . . . . . . . . . . . 12 |
32 | elxr 9563 | . . . . . . . . . . . . . 14 | |
33 | 10, 32 | sylib 121 | . . . . . . . . . . . . 13 |
34 | 33 | ad4antr 485 | . . . . . . . . . . . 12 |
35 | 31, 29, 34 | ecase23d 1328 | . . . . . . . . . . 11 |
36 | 4 | ad3antrrr 483 | . . . . . . . . . . . 12 |
37 | 15 | ad2antrr 479 | . . . . . . . . . . . 12 |
38 | elxr 9563 | . . . . . . . . . . . . . 14 | |
39 | 20, 38 | sylib 121 | . . . . . . . . . . . . 13 |
40 | 39 | ad4antr 485 | . . . . . . . . . . . 12 |
41 | 36, 37, 40 | ecase23d 1328 | . . . . . . . . . . 11 |
42 | 35, 41 | jca 304 | . . . . . . . . . 10 |
43 | 6 | ad4antr 485 | . . . . . . . . . . 11 |
44 | 35, 41, 43 | ltled 7881 | . . . . . . . . . 10 |
45 | maxleim 10977 | . . . . . . . . . 10 | |
46 | 42, 44, 45 | sylc 62 | . . . . . . . . 9 |
47 | 30, 46 | eqtrd 2172 | . . . . . . . 8 |
48 | xrmnfdc 9626 | . . . . . . . . . 10 DECID | |
49 | exmiddc 821 | . . . . . . . . . 10 DECID | |
50 | 10, 48, 49 | 3syl 17 | . . . . . . . . 9 |
51 | 50 | ad3antrrr 483 | . . . . . . . 8 |
52 | 28, 47, 51 | mpjaodan 787 | . . . . . . 7 |
53 | 26, 52 | eqtrd 2172 | . . . . . 6 |
54 | xrpnfdc 9625 | . . . . . . . 8 DECID | |
55 | exmiddc 821 | . . . . . . . 8 DECID | |
56 | 10, 54, 55 | 3syl 17 | . . . . . . 7 |
57 | 56 | ad2antrr 479 | . . . . . 6 |
58 | 24, 53, 57 | mpjaodan 787 | . . . . 5 |
59 | 16, 58 | eqtrd 2172 | . . . 4 |
60 | xrmnfdc 9626 | . . . . . 6 DECID | |
61 | exmiddc 821 | . . . . . 6 DECID | |
62 | 20, 60, 61 | 3syl 17 | . . . . 5 |
63 | 62 | adantr 274 | . . . 4 |
64 | 14, 59, 63 | mpjaodan 787 | . . 3 |
65 | 5, 64 | eqtrd 2172 | . 2 |
66 | xrpnfdc 9625 | . . 3 DECID | |
67 | exmiddc 821 | . . 3 DECID | |
68 | 20, 66, 67 | 3syl 17 | . 2 |
69 | 3, 65, 68 | mpjaodan 787 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 w3o 961 wceq 1331 wcel 1480 cif 3474 cpr 3528 class class class wbr 3929 csup 6869 cr 7619 cpnf 7797 cmnf 7798 cxr 7799 clt 7800 cle 7801 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-apti 7735 |
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-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-iota 5088 df-riota 5730 df-sup 6871 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 |
This theorem is referenced by: xrmaxiflemlub 11017 |
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