Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xleadd1a | Unicode version |
Description: Extended real version of leadd1 8319; note that the converse implication is not true, unlike the real version (for example but ). (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xleadd1a |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplrr 526 | . . . . . . 7 | |
2 | simpr 109 | . . . . . . 7 | |
3 | simplrl 525 | . . . . . . 7 | |
4 | simpllr 524 | . . . . . . 7 | |
5 | 1, 2, 3, 4 | leadd1dd 8448 | . . . . . 6 |
6 | 1, 3 | rexaddd 9781 | . . . . . 6 |
7 | 2, 3 | rexaddd 9781 | . . . . . 6 |
8 | 5, 6, 7 | 3brtr4d 4008 | . . . . 5 |
9 | simpl1 989 | . . . . . . . . 9 | |
10 | simpl3 991 | . . . . . . . . 9 | |
11 | xaddcl 9787 | . . . . . . . . 9 | |
12 | 9, 10, 11 | syl2anc 409 | . . . . . . . 8 |
13 | 12 | ad2antrr 480 | . . . . . . 7 |
14 | pnfge 9716 | . . . . . . 7 | |
15 | 13, 14 | syl 14 | . . . . . 6 |
16 | oveq1 5843 | . . . . . . 7 | |
17 | rexr 7935 | . . . . . . . . 9 | |
18 | renemnf 7938 | . . . . . . . . 9 | |
19 | xaddpnf2 9774 | . . . . . . . . 9 | |
20 | 17, 18, 19 | syl2anc 409 | . . . . . . . 8 |
21 | 20 | ad2antrl 482 | . . . . . . 7 |
22 | 16, 21 | sylan9eqr 2219 | . . . . . 6 |
23 | 15, 22 | breqtrrd 4004 | . . . . 5 |
24 | 12 | adantr 274 | . . . . . . . 8 |
25 | 24 | xrleidd 9728 | . . . . . . 7 |
26 | simplr 520 | . . . . . . . . 9 | |
27 | simpr 109 | . . . . . . . . . 10 | |
28 | 9 | adantr 274 | . . . . . . . . . . 11 |
29 | mnfle 9719 | . . . . . . . . . . 11 | |
30 | 28, 29 | syl 14 | . . . . . . . . . 10 |
31 | 27, 30 | eqbrtrd 3998 | . . . . . . . . 9 |
32 | simpl2 990 | . . . . . . . . . . 11 | |
33 | xrletri3 9731 | . . . . . . . . . . 11 | |
34 | 9, 32, 33 | syl2anc 409 | . . . . . . . . . 10 |
35 | 34 | adantr 274 | . . . . . . . . 9 |
36 | 26, 31, 35 | mpbir2and 933 | . . . . . . . 8 |
37 | 36 | oveq1d 5851 | . . . . . . 7 |
38 | 25, 37 | breqtrd 4002 | . . . . . 6 |
39 | 38 | adantlr 469 | . . . . 5 |
40 | elxr 9703 | . . . . . . 7 | |
41 | 32, 40 | sylib 121 | . . . . . 6 |
42 | 41 | adantr 274 | . . . . 5 |
43 | 8, 23, 39, 42 | mpjao3dan 1296 | . . . 4 |
44 | 43 | anassrs 398 | . . 3 |
45 | 12 | adantr 274 | . . . . . 6 |
46 | 45 | xrleidd 9728 | . . . . 5 |
47 | simplr 520 | . . . . . . 7 | |
48 | pnfge 9716 | . . . . . . . . . 10 | |
49 | 32, 48 | syl 14 | . . . . . . . . 9 |
50 | 49 | adantr 274 | . . . . . . . 8 |
51 | simpr 109 | . . . . . . . 8 | |
52 | 50, 51 | breqtrrd 4004 | . . . . . . 7 |
53 | 34 | adantr 274 | . . . . . . 7 |
54 | 47, 52, 53 | mpbir2and 933 | . . . . . 6 |
55 | 54 | oveq1d 5851 | . . . . 5 |
56 | 46, 55 | breqtrd 4002 | . . . 4 |
57 | 56 | adantlr 469 | . . 3 |
58 | oveq1 5843 | . . . . 5 | |
59 | renepnf 7937 | . . . . . . 7 | |
60 | xaddmnf2 9776 | . . . . . . 7 | |
61 | 17, 59, 60 | syl2anc 409 | . . . . . 6 |
62 | 61 | adantl 275 | . . . . 5 |
63 | 58, 62 | sylan9eqr 2219 | . . . 4 |
64 | xaddcl 9787 | . . . . . . 7 | |
65 | 32, 10, 64 | syl2anc 409 | . . . . . 6 |
66 | 65 | ad2antrr 480 | . . . . 5 |
67 | mnfle 9719 | . . . . 5 | |
68 | 66, 67 | syl 14 | . . . 4 |
69 | 63, 68 | eqbrtrd 3998 | . . 3 |
70 | elxr 9703 | . . . . 5 | |
71 | 9, 70 | sylib 121 | . . . 4 |
72 | 71 | adantr 274 | . . 3 |
73 | 44, 57, 69, 72 | mpjao3dan 1296 | . 2 |
74 | 38 | adantlr 469 | . . 3 |
75 | 12 | ad2antrr 480 | . . . . 5 |
76 | 75, 14 | syl 14 | . . . 4 |
77 | simplr 520 | . . . . . 6 | |
78 | 77 | oveq2d 5852 | . . . . 5 |
79 | 32 | adantr 274 | . . . . . 6 |
80 | xaddpnf1 9773 | . . . . . 6 | |
81 | 79, 80 | sylan 281 | . . . . 5 |
82 | 78, 81 | eqtrd 2197 | . . . 4 |
83 | 76, 82 | breqtrrd 4004 | . . 3 |
84 | xrmnfdc 9770 | . . . . . 6 DECID | |
85 | exmiddc 826 | . . . . . 6 DECID | |
86 | 84, 85 | syl 14 | . . . . 5 |
87 | df-ne 2335 | . . . . . 6 | |
88 | 87 | orbi2i 752 | . . . . 5 |
89 | 86, 88 | sylibr 133 | . . . 4 |
90 | 79, 89 | syl 14 | . . 3 |
91 | 74, 83, 90 | mpjaodan 788 | . 2 |
92 | 56 | adantlr 469 | . . 3 |
93 | simplr 520 | . . . . . 6 | |
94 | 93 | oveq2d 5852 | . . . . 5 |
95 | 9 | adantr 274 | . . . . . 6 |
96 | xaddmnf1 9775 | . . . . . 6 | |
97 | 95, 96 | sylan 281 | . . . . 5 |
98 | 94, 97 | eqtrd 2197 | . . . 4 |
99 | 65 | ad2antrr 480 | . . . . 5 |
100 | 99, 67 | syl 14 | . . . 4 |
101 | 98, 100 | eqbrtrd 3998 | . . 3 |
102 | xrpnfdc 9769 | . . . . . 6 DECID | |
103 | exmiddc 826 | . . . . . 6 DECID | |
104 | 102, 103 | syl 14 | . . . . 5 |
105 | df-ne 2335 | . . . . . 6 | |
106 | 105 | orbi2i 752 | . . . . 5 |
107 | 104, 106 | sylibr 133 | . . . 4 |
108 | 95, 107 | syl 14 | . . 3 |
109 | 92, 101, 108 | mpjaodan 788 | . 2 |
110 | elxr 9703 | . . 3 | |
111 | 10, 110 | sylib 121 | . 2 |
112 | 73, 91, 109, 111 | mpjao3dan 1296 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 824 w3o 966 w3a 967 wceq 1342 wcel 2135 wne 2334 class class class wbr 3976 (class class class)co 5836 cr 7743 caddc 7747 cpnf 7921 cmnf 7922 cxr 7923 cle 7925 cxad 9697 |
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-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-apti 7859 ax-pre-ltadd 7860 |
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-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 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-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 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-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-xadd 9700 |
This theorem is referenced by: xleadd2a 9801 xleadd1 9802 xaddge0 9805 xle2add 9806 xblss2ps 12945 xblss2 12946 |
Copyright terms: Public domain | W3C validator |