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Mirrors > Home > ILE Home > Th. List > xaddval | Unicode version |
Description: Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7805 | . . . . . 6 | |
2 | 1 | a1i 9 | . . . . 5 |
3 | pnfxr 7811 | . . . . . 6 | |
4 | 3 | a1i 9 | . . . . 5 |
5 | xrmnfdc 9619 | . . . . . 6 DECID | |
6 | 5 | adantl 275 | . . . . 5 DECID |
7 | 2, 4, 6 | ifcldcd 3502 | . . . 4 |
8 | 7 | adantr 274 | . . 3 |
9 | mnfxr 7815 | . . . . . . 7 | |
10 | 9 | a1i 9 | . . . . . 6 |
11 | xrpnfdc 9618 | . . . . . . 7 DECID | |
12 | 11 | adantl 275 | . . . . . 6 DECID |
13 | 2, 10, 12 | ifcldcd 3502 | . . . . 5 |
14 | 13 | ad2antrr 479 | . . . 4 |
15 | 3 | a1i 9 | . . . . 5 |
16 | 9 | a1i 9 | . . . . . 6 |
17 | simp-4r 531 | . . . . . . . . 9 | |
18 | simpl 108 | . . . . . . . . . . 11 | |
19 | 18 | ad4antr 485 | . . . . . . . . . 10 |
20 | simpllr 523 | . . . . . . . . . . 11 | |
21 | 20 | neqned 2313 | . . . . . . . . . 10 |
22 | xrnemnf 9557 | . . . . . . . . . . 11 | |
23 | 22 | biimpi 119 | . . . . . . . . . 10 |
24 | 19, 21, 23 | syl2anc 408 | . . . . . . . . 9 |
25 | 17, 24 | ecased 1327 | . . . . . . . 8 |
26 | simplr 519 | . . . . . . . . 9 | |
27 | simpr 109 | . . . . . . . . . . 11 | |
28 | 27 | ad4antr 485 | . . . . . . . . . 10 |
29 | simpr 109 | . . . . . . . . . . 11 | |
30 | 29 | neqned 2313 | . . . . . . . . . 10 |
31 | xrnemnf 9557 | . . . . . . . . . . 11 | |
32 | 31 | biimpi 119 | . . . . . . . . . 10 |
33 | 28, 30, 32 | syl2anc 408 | . . . . . . . . 9 |
34 | 26, 33 | ecased 1327 | . . . . . . . 8 |
35 | 25, 34 | readdcld 7788 | . . . . . . 7 |
36 | 35 | rexrd 7808 | . . . . . 6 |
37 | 6 | ad3antrrr 483 | . . . . . 6 DECID |
38 | 16, 36, 37 | ifcldadc 3496 | . . . . 5 |
39 | 12 | ad2antrr 479 | . . . . 5 DECID |
40 | 15, 38, 39 | ifcldadc 3496 | . . . 4 |
41 | xrmnfdc 9619 | . . . . 5 DECID | |
42 | 41 | ad2antrr 479 | . . . 4 DECID |
43 | 14, 40, 42 | ifcldadc 3496 | . . 3 |
44 | xrpnfdc 9618 | . . . 4 DECID | |
45 | 44 | adantr 274 | . . 3 DECID |
46 | 8, 43, 45 | ifcldadc 3496 | . 2 |
47 | simpl 108 | . . . . 5 | |
48 | 47 | eqeq1d 2146 | . . . 4 |
49 | simpr 109 | . . . . . 6 | |
50 | 49 | eqeq1d 2146 | . . . . 5 |
51 | 50 | ifbid 3488 | . . . 4 |
52 | 47 | eqeq1d 2146 | . . . . 5 |
53 | 49 | eqeq1d 2146 | . . . . . 6 |
54 | 53 | ifbid 3488 | . . . . 5 |
55 | oveq12 5776 | . . . . . . 7 | |
56 | 50, 55 | ifbieq2d 3491 | . . . . . 6 |
57 | 53, 56 | ifbieq2d 3491 | . . . . 5 |
58 | 52, 54, 57 | ifbieq12d 3493 | . . . 4 |
59 | 48, 51, 58 | ifbieq12d 3493 | . . 3 |
60 | df-xadd 9553 | . . 3 | |
61 | 59, 60 | ovmpoga 5893 | . 2 |
62 | 46, 61 | mpd3an3 1316 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 wceq 1331 wcel 1480 wne 2306 cif 3469 (class class class)co 5767 cr 7612 cc0 7613 caddc 7616 cpnf 7790 cmnf 7791 cxr 7792 cxad 9550 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-rnegex 7722 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-xadd 9553 |
This theorem is referenced by: xaddpnf1 9622 xaddpnf2 9623 xaddmnf1 9624 xaddmnf2 9625 pnfaddmnf 9626 mnfaddpnf 9627 rexadd 9628 |
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