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Mirrors > Home > ILE Home > Th. List > xltadd1 | Unicode version |
Description: Extended real version of ltadd1 8194. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.) |
Ref | Expression |
---|---|
xltadd1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 519 | . . . 4 | |
2 | simpr 109 | . . . 4 | |
3 | simpll3 1022 | . . . 4 | |
4 | ltadd1 8194 | . . . . 5 | |
5 | simp1 981 | . . . . . . 7 | |
6 | simp3 983 | . . . . . . 7 | |
7 | 5, 6 | rexaddd 9640 | . . . . . 6 |
8 | simp2 982 | . . . . . . 7 | |
9 | 8, 6 | rexaddd 9640 | . . . . . 6 |
10 | 7, 9 | breq12d 3942 | . . . . 5 |
11 | 4, 10 | bitr4d 190 | . . . 4 |
12 | 1, 2, 3, 11 | syl3anc 1216 | . . 3 |
13 | ltpnf 9570 | . . . . . 6 | |
14 | 13 | ad2antlr 480 | . . . . 5 |
15 | breq2 3933 | . . . . . 6 | |
16 | 15 | adantl 275 | . . . . 5 |
17 | 14, 16 | mpbird 166 | . . . 4 |
18 | simplr 519 | . . . . . . 7 | |
19 | simpll3 1022 | . . . . . . 7 | |
20 | rexadd 9638 | . . . . . . . 8 | |
21 | readdcl 7749 | . . . . . . . 8 | |
22 | 20, 21 | eqeltrd 2216 | . . . . . . 7 |
23 | 18, 19, 22 | syl2anc 408 | . . . . . 6 |
24 | ltpnf 9570 | . . . . . 6 | |
25 | 23, 24 | syl 14 | . . . . 5 |
26 | oveq1 5781 | . . . . . . 7 | |
27 | 26 | adantl 275 | . . . . . 6 |
28 | rexr 7814 | . . . . . . . 8 | |
29 | renemnf 7817 | . . . . . . . 8 | |
30 | xaddpnf2 9633 | . . . . . . . 8 | |
31 | 28, 29, 30 | syl2anc 408 | . . . . . . 7 |
32 | 19, 31 | syl 14 | . . . . . 6 |
33 | 27, 32 | eqtrd 2172 | . . . . 5 |
34 | 25, 33 | breqtrrd 3956 | . . . 4 |
35 | 17, 34 | 2thd 174 | . . 3 |
36 | mnfle 9581 | . . . . . . . 8 | |
37 | 36 | 3ad2ant1 1002 | . . . . . . 7 |
38 | 37 | ad2antrr 479 | . . . . . 6 |
39 | mnfxr 7825 | . . . . . . 7 | |
40 | simpll1 1020 | . . . . . . 7 | |
41 | xrlenlt 7832 | . . . . . . 7 | |
42 | 39, 40, 41 | sylancr 410 | . . . . . 6 |
43 | 38, 42 | mpbid 146 | . . . . 5 |
44 | breq2 3933 | . . . . . 6 | |
45 | 44 | adantl 275 | . . . . 5 |
46 | 43, 45 | mtbird 662 | . . . 4 |
47 | 28 | 3ad2ant3 1004 | . . . . . . . . 9 |
48 | 47 | ad2antrr 479 | . . . . . . . 8 |
49 | xaddcl 9646 | . . . . . . . 8 | |
50 | 40, 48, 49 | syl2anc 408 | . . . . . . 7 |
51 | mnfle 9581 | . . . . . . 7 | |
52 | 50, 51 | syl 14 | . . . . . 6 |
53 | xrlenlt 7832 | . . . . . . 7 | |
54 | 39, 50, 53 | sylancr 410 | . . . . . 6 |
55 | 52, 54 | mpbid 146 | . . . . 5 |
56 | simpr 109 | . . . . . . . 8 | |
57 | 56 | oveq1d 5789 | . . . . . . 7 |
58 | renepnf 7816 | . . . . . . . . . 10 | |
59 | 58 | 3ad2ant3 1004 | . . . . . . . . 9 |
60 | 59 | ad2antrr 479 | . . . . . . . 8 |
61 | xaddmnf2 9635 | . . . . . . . 8 | |
62 | 48, 60, 61 | syl2anc 408 | . . . . . . 7 |
63 | 57, 62 | eqtrd 2172 | . . . . . 6 |
64 | 63 | breq2d 3941 | . . . . 5 |
65 | 55, 64 | mtbird 662 | . . . 4 |
66 | 46, 65 | 2falsed 691 | . . 3 |
67 | elxr 9566 | . . . . . 6 | |
68 | 67 | biimpi 119 | . . . . 5 |
69 | 68 | 3ad2ant2 1003 | . . . 4 |
70 | 69 | adantr 274 | . . 3 |
71 | 12, 35, 66, 70 | mpjao3dan 1285 | . 2 |
72 | simpl2 985 | . . . . . 6 | |
73 | pnfge 9578 | . . . . . 6 | |
74 | 72, 73 | syl 14 | . . . . 5 |
75 | pnfxr 7821 | . . . . . . 7 | |
76 | 75 | a1i 9 | . . . . . 6 |
77 | xrlenlt 7832 | . . . . . 6 | |
78 | 72, 76, 77 | syl2anc 408 | . . . . 5 |
79 | 74, 78 | mpbid 146 | . . . 4 |
80 | simpr 109 | . . . . 5 | |
81 | 80 | breq1d 3939 | . . . 4 |
82 | 79, 81 | mtbird 662 | . . 3 |
83 | 47 | adantr 274 | . . . . . . . 8 |
84 | xaddcl 9646 | . . . . . . . 8 | |
85 | 72, 83, 84 | syl2anc 408 | . . . . . . 7 |
86 | pnfge 9578 | . . . . . . 7 | |
87 | 85, 86 | syl 14 | . . . . . 6 |
88 | 29 | 3ad2ant3 1004 | . . . . . . . 8 |
89 | 88 | adantr 274 | . . . . . . 7 |
90 | 83, 89, 30 | syl2anc 408 | . . . . . 6 |
91 | 87, 90 | breqtrrd 3956 | . . . . 5 |
92 | xaddcl 9646 | . . . . . . 7 | |
93 | 75, 83, 92 | sylancr 410 | . . . . . 6 |
94 | xrlenlt 7832 | . . . . . 6 | |
95 | 85, 93, 94 | syl2anc 408 | . . . . 5 |
96 | 91, 95 | mpbid 146 | . . . 4 |
97 | 80 | oveq1d 5789 | . . . . 5 |
98 | 97 | breq1d 3939 | . . . 4 |
99 | 96, 98 | mtbird 662 | . . 3 |
100 | 82, 99 | 2falsed 691 | . 2 |
101 | simplr 519 | . . . . 5 | |
102 | mnflt 9572 | . . . . . 6 | |
103 | 102 | adantl 275 | . . . . 5 |
104 | 101, 103 | eqbrtrd 3950 | . . . 4 |
105 | 101 | oveq1d 5789 | . . . . . 6 |
106 | simpll3 1022 | . . . . . . . 8 | |
107 | 106, 28 | syl 14 | . . . . . . 7 |
108 | 106, 58 | syl 14 | . . . . . . 7 |
109 | 107, 108, 61 | syl2anc 408 | . . . . . 6 |
110 | 105, 109 | eqtrd 2172 | . . . . 5 |
111 | simpr 109 | . . . . . . 7 | |
112 | rexadd 9638 | . . . . . . . 8 | |
113 | readdcl 7749 | . . . . . . . 8 | |
114 | 112, 113 | eqeltrd 2216 | . . . . . . 7 |
115 | 111, 106, 114 | syl2anc 408 | . . . . . 6 |
116 | mnflt 9572 | . . . . . 6 | |
117 | 115, 116 | syl 14 | . . . . 5 |
118 | 110, 117 | eqbrtrd 3950 | . . . 4 |
119 | 104, 118 | 2thd 174 | . . 3 |
120 | simplr 519 | . . . . 5 | |
121 | simpr 109 | . . . . 5 | |
122 | 120, 121 | breq12d 3942 | . . . 4 |
123 | oveq1 5781 | . . . . . . 7 | |
124 | 47, 59, 61 | syl2anc 408 | . . . . . . 7 |
125 | 123, 124 | sylan9eqr 2194 | . . . . . 6 |
126 | 125 | adantr 274 | . . . . 5 |
127 | 26 | adantl 275 | . . . . . 6 |
128 | 47, 88, 30 | syl2anc 408 | . . . . . . 7 |
129 | 128 | ad2antrr 479 | . . . . . 6 |
130 | 127, 129 | eqtrd 2172 | . . . . 5 |
131 | 126, 130 | breq12d 3942 | . . . 4 |
132 | 122, 131 | bitr4d 190 | . . 3 |
133 | simplr 519 | . . . . 5 | |
134 | simpr 109 | . . . . 5 | |
135 | 133, 134 | breq12d 3942 | . . . 4 |
136 | 124 | ad2antrr 479 | . . . . . 6 |
137 | 123 | eqeq1d 2148 | . . . . . . 7 |
138 | 137 | ad2antlr 480 | . . . . . 6 |
139 | 136, 138 | mpbird 166 | . . . . 5 |
140 | 134 | oveq1d 5789 | . . . . . 6 |
141 | 140, 136 | eqtrd 2172 | . . . . 5 |
142 | 139, 141 | breq12d 3942 | . . . 4 |
143 | 135, 142 | bitr4d 190 | . . 3 |
144 | 69 | adantr 274 | . . 3 |
145 | 119, 132, 143, 144 | mpjao3dan 1285 | . 2 |
146 | elxr 9566 | . . . 4 | |
147 | 146 | biimpi 119 | . . 3 |
148 | 147 | 3ad2ant1 1002 | . 2 |
149 | 71, 100, 145, 148 | mpjao3dan 1285 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 961 w3a 962 wceq 1331 wcel 1480 wne 2308 class class class wbr 3929 (class class class)co 5774 cr 7622 caddc 7626 cpnf 7800 cmnf 7801 cxr 7802 clt 7803 cle 7804 cxad 9560 |
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 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-i2m1 7728 ax-0id 7731 ax-rnegex 7732 ax-pre-ltadd 7739 |
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-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 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-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 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-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-xadd 9563 |
This theorem is referenced by: xltadd2 9663 xlt2add 9666 xrmaxaddlem 11032 |
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