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Mirrors > Home > ILE Home > Th. List > xaddcom | Unicode version |
Description: The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.) |
Ref | Expression |
---|---|
xaddcom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9570 | . 2 | |
2 | elxr 9570 | . . . 4 | |
3 | recn 7760 | . . . . . . 7 | |
4 | recn 7760 | . . . . . . 7 | |
5 | addcom 7906 | . . . . . . 7 | |
6 | 3, 4, 5 | syl2an 287 | . . . . . 6 |
7 | rexadd 9642 | . . . . . 6 | |
8 | rexadd 9642 | . . . . . . 7 | |
9 | 8 | ancoms 266 | . . . . . 6 |
10 | 6, 7, 9 | 3eqtr4d 2182 | . . . . 5 |
11 | oveq2 5782 | . . . . . . 7 | |
12 | rexr 7818 | . . . . . . . 8 | |
13 | renemnf 7821 | . . . . . . . 8 | |
14 | xaddpnf1 9636 | . . . . . . . 8 | |
15 | 12, 13, 14 | syl2anc 408 | . . . . . . 7 |
16 | 11, 15 | sylan9eqr 2194 | . . . . . 6 |
17 | oveq1 5781 | . . . . . . 7 | |
18 | xaddpnf2 9637 | . . . . . . . 8 | |
19 | 12, 13, 18 | syl2anc 408 | . . . . . . 7 |
20 | 17, 19 | sylan9eqr 2194 | . . . . . 6 |
21 | 16, 20 | eqtr4d 2175 | . . . . 5 |
22 | oveq2 5782 | . . . . . . 7 | |
23 | renepnf 7820 | . . . . . . . 8 | |
24 | xaddmnf1 9638 | . . . . . . . 8 | |
25 | 12, 23, 24 | syl2anc 408 | . . . . . . 7 |
26 | 22, 25 | sylan9eqr 2194 | . . . . . 6 |
27 | oveq1 5781 | . . . . . . 7 | |
28 | xaddmnf2 9639 | . . . . . . . 8 | |
29 | 12, 23, 28 | syl2anc 408 | . . . . . . 7 |
30 | 27, 29 | sylan9eqr 2194 | . . . . . 6 |
31 | 26, 30 | eqtr4d 2175 | . . . . 5 |
32 | 10, 21, 31 | 3jaodan 1284 | . . . 4 |
33 | 2, 32 | sylan2b 285 | . . 3 |
34 | pnfaddmnf 9640 | . . . . . . . 8 | |
35 | mnfaddpnf 9641 | . . . . . . . 8 | |
36 | 34, 35 | eqtr4i 2163 | . . . . . . 7 |
37 | simpr 109 | . . . . . . . 8 | |
38 | 37 | oveq2d 5790 | . . . . . . 7 |
39 | 37 | oveq1d 5789 | . . . . . . 7 |
40 | 36, 38, 39 | 3eqtr4a 2198 | . . . . . 6 |
41 | xaddpnf2 9637 | . . . . . . 7 | |
42 | xaddpnf1 9636 | . . . . . . 7 | |
43 | 41, 42 | eqtr4d 2175 | . . . . . 6 |
44 | xrmnfdc 9633 | . . . . . . . 8 DECID | |
45 | exmiddc 821 | . . . . . . . 8 DECID | |
46 | 44, 45 | syl 14 | . . . . . . 7 |
47 | df-ne 2309 | . . . . . . . 8 | |
48 | 47 | orbi2i 751 | . . . . . . 7 |
49 | 46, 48 | sylibr 133 | . . . . . 6 |
50 | 40, 43, 49 | mpjaodan 787 | . . . . 5 |
51 | 50 | adantl 275 | . . . 4 |
52 | simpl 108 | . . . . 5 | |
53 | 52 | oveq1d 5789 | . . . 4 |
54 | 52 | oveq2d 5790 | . . . 4 |
55 | 51, 53, 54 | 3eqtr4d 2182 | . . 3 |
56 | 35, 34 | eqtr4i 2163 | . . . . . . 7 |
57 | simpr 109 | . . . . . . . 8 | |
58 | 57 | oveq2d 5790 | . . . . . . 7 |
59 | 57 | oveq1d 5789 | . . . . . . 7 |
60 | 56, 58, 59 | 3eqtr4a 2198 | . . . . . 6 |
61 | xaddmnf2 9639 | . . . . . . 7 | |
62 | xaddmnf1 9638 | . . . . . . 7 | |
63 | 61, 62 | eqtr4d 2175 | . . . . . 6 |
64 | xrpnfdc 9632 | . . . . . . . 8 DECID | |
65 | exmiddc 821 | . . . . . . . 8 DECID | |
66 | 64, 65 | syl 14 | . . . . . . 7 |
67 | df-ne 2309 | . . . . . . . 8 | |
68 | 67 | orbi2i 751 | . . . . . . 7 |
69 | 66, 68 | sylibr 133 | . . . . . 6 |
70 | 60, 63, 69 | mpjaodan 787 | . . . . 5 |
71 | 70 | adantl 275 | . . . 4 |
72 | simpl 108 | . . . . 5 | |
73 | 72 | oveq1d 5789 | . . . 4 |
74 | 72 | oveq2d 5790 | . . . 4 |
75 | 71, 73, 74 | 3eqtr4d 2182 | . . 3 |
76 | 33, 55, 75 | 3jaoian 1283 | . 2 |
77 | 1, 76 | sylanb 282 | 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 wne 2308 (class class class)co 5774 cc 7625 cr 7626 cc0 7627 caddc 7630 cpnf 7804 cmnf 7805 cxr 7806 cxad 9564 |
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 7718 ax-resscn 7719 ax-1re 7721 ax-addrcl 7724 ax-addcom 7727 ax-rnegex 7736 |
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-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-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7809 df-mnf 7810 df-xr 7811 df-xadd 9567 |
This theorem is referenced by: xaddid2 9653 xleadd2a 9664 xltadd2 9667 xadd4d 9675 xrmaxaddlem 11036 |
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