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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 9703 | . 2 | |
2 | elxr 9703 | . . . 4 | |
3 | recn 7877 | . . . . . . 7 | |
4 | recn 7877 | . . . . . . 7 | |
5 | addcom 8026 | . . . . . . 7 | |
6 | 3, 4, 5 | syl2an 287 | . . . . . 6 |
7 | rexadd 9779 | . . . . . 6 | |
8 | rexadd 9779 | . . . . . . 7 | |
9 | 8 | ancoms 266 | . . . . . 6 |
10 | 6, 7, 9 | 3eqtr4d 2207 | . . . . 5 |
11 | oveq2 5844 | . . . . . . 7 | |
12 | rexr 7935 | . . . . . . . 8 | |
13 | renemnf 7938 | . . . . . . . 8 | |
14 | xaddpnf1 9773 | . . . . . . . 8 | |
15 | 12, 13, 14 | syl2anc 409 | . . . . . . 7 |
16 | 11, 15 | sylan9eqr 2219 | . . . . . 6 |
17 | oveq1 5843 | . . . . . . 7 | |
18 | xaddpnf2 9774 | . . . . . . . 8 | |
19 | 12, 13, 18 | syl2anc 409 | . . . . . . 7 |
20 | 17, 19 | sylan9eqr 2219 | . . . . . 6 |
21 | 16, 20 | eqtr4d 2200 | . . . . 5 |
22 | oveq2 5844 | . . . . . . 7 | |
23 | renepnf 7937 | . . . . . . . 8 | |
24 | xaddmnf1 9775 | . . . . . . . 8 | |
25 | 12, 23, 24 | syl2anc 409 | . . . . . . 7 |
26 | 22, 25 | sylan9eqr 2219 | . . . . . 6 |
27 | oveq1 5843 | . . . . . . 7 | |
28 | xaddmnf2 9776 | . . . . . . . 8 | |
29 | 12, 23, 28 | syl2anc 409 | . . . . . . 7 |
30 | 27, 29 | sylan9eqr 2219 | . . . . . 6 |
31 | 26, 30 | eqtr4d 2200 | . . . . 5 |
32 | 10, 21, 31 | 3jaodan 1295 | . . . 4 |
33 | 2, 32 | sylan2b 285 | . . 3 |
34 | pnfaddmnf 9777 | . . . . . . . 8 | |
35 | mnfaddpnf 9778 | . . . . . . . 8 | |
36 | 34, 35 | eqtr4i 2188 | . . . . . . 7 |
37 | simpr 109 | . . . . . . . 8 | |
38 | 37 | oveq2d 5852 | . . . . . . 7 |
39 | 37 | oveq1d 5851 | . . . . . . 7 |
40 | 36, 38, 39 | 3eqtr4a 2223 | . . . . . 6 |
41 | xaddpnf2 9774 | . . . . . . 7 | |
42 | xaddpnf1 9773 | . . . . . . 7 | |
43 | 41, 42 | eqtr4d 2200 | . . . . . 6 |
44 | xrmnfdc 9770 | . . . . . . . 8 DECID | |
45 | exmiddc 826 | . . . . . . . 8 DECID | |
46 | 44, 45 | syl 14 | . . . . . . 7 |
47 | df-ne 2335 | . . . . . . . 8 | |
48 | 47 | orbi2i 752 | . . . . . . 7 |
49 | 46, 48 | sylibr 133 | . . . . . 6 |
50 | 40, 43, 49 | mpjaodan 788 | . . . . 5 |
51 | 50 | adantl 275 | . . . 4 |
52 | simpl 108 | . . . . 5 | |
53 | 52 | oveq1d 5851 | . . . 4 |
54 | 52 | oveq2d 5852 | . . . 4 |
55 | 51, 53, 54 | 3eqtr4d 2207 | . . 3 |
56 | 35, 34 | eqtr4i 2188 | . . . . . . 7 |
57 | simpr 109 | . . . . . . . 8 | |
58 | 57 | oveq2d 5852 | . . . . . . 7 |
59 | 57 | oveq1d 5851 | . . . . . . 7 |
60 | 56, 58, 59 | 3eqtr4a 2223 | . . . . . 6 |
61 | xaddmnf2 9776 | . . . . . . 7 | |
62 | xaddmnf1 9775 | . . . . . . 7 | |
63 | 61, 62 | eqtr4d 2200 | . . . . . 6 |
64 | xrpnfdc 9769 | . . . . . . . 8 DECID | |
65 | exmiddc 826 | . . . . . . . 8 DECID | |
66 | 64, 65 | syl 14 | . . . . . . 7 |
67 | df-ne 2335 | . . . . . . . 8 | |
68 | 67 | orbi2i 752 | . . . . . . 7 |
69 | 66, 68 | sylibr 133 | . . . . . 6 |
70 | 60, 63, 69 | mpjaodan 788 | . . . . 5 |
71 | 70 | adantl 275 | . . . 4 |
72 | simpl 108 | . . . . 5 | |
73 | 72 | oveq1d 5851 | . . . 4 |
74 | 72 | oveq2d 5852 | . . . 4 |
75 | 71, 73, 74 | 3eqtr4d 2207 | . . 3 |
76 | 33, 55, 75 | 3jaoian 1294 | . 2 |
77 | 1, 76 | sylanb 282 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 824 w3o 966 wceq 1342 wcel 2135 wne 2334 (class class class)co 5836 cc 7742 cr 7743 cc0 7744 caddc 7747 cpnf 7921 cmnf 7922 cxr 7923 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-1re 7838 ax-addrcl 7841 ax-addcom 7844 ax-rnegex 7853 |
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-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-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-xadd 9700 |
This theorem is referenced by: xaddid2 9790 xleadd2a 9801 xltadd2 9804 xadd4d 9812 xrmaxaddlem 11187 |
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