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Mirrors > Home > ILE Home > Th. List > xaddass | Unicode version |
Description: Associativity of extended real addition. The correct condition here is "it is not the case that both and appear as one of , i.e. ", but this condition is difficult to work with, so we break the theorem into two parts: this one, where is not present in , and xaddass2 9653, where is not present. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 7753 | . . . . . . . . . 10 | |
2 | recn 7753 | . . . . . . . . . 10 | |
3 | recn 7753 | . . . . . . . . . 10 | |
4 | addass 7750 | . . . . . . . . . 10 | |
5 | 1, 2, 3, 4 | syl3an 1258 | . . . . . . . . 9 |
6 | 5 | 3expa 1181 | . . . . . . . 8 |
7 | readdcl 7746 | . . . . . . . . 9 | |
8 | rexadd 9635 | . . . . . . . . 9 | |
9 | 7, 8 | sylan 281 | . . . . . . . 8 |
10 | readdcl 7746 | . . . . . . . . . 10 | |
11 | rexadd 9635 | . . . . . . . . . 10 | |
12 | 10, 11 | sylan2 284 | . . . . . . . . 9 |
13 | 12 | anassrs 397 | . . . . . . . 8 |
14 | 6, 9, 13 | 3eqtr4d 2182 | . . . . . . 7 |
15 | rexadd 9635 | . . . . . . . . 9 | |
16 | 15 | adantr 274 | . . . . . . . 8 |
17 | 16 | oveq1d 5789 | . . . . . . 7 |
18 | rexadd 9635 | . . . . . . . . 9 | |
19 | 18 | adantll 467 | . . . . . . . 8 |
20 | 19 | oveq2d 5790 | . . . . . . 7 |
21 | 14, 17, 20 | 3eqtr4d 2182 | . . . . . 6 |
22 | 21 | adantll 467 | . . . . 5 |
23 | oveq2 5782 | . . . . . . . . 9 | |
24 | simp1l 1005 | . . . . . . . . . . 11 | |
25 | simp2l 1007 | . . . . . . . . . . 11 | |
26 | xaddcl 9643 | . . . . . . . . . . 11 | |
27 | 24, 25, 26 | syl2anc 408 | . . . . . . . . . 10 |
28 | xaddnemnf 9640 | . . . . . . . . . . 11 | |
29 | 28 | 3adant3 1001 | . . . . . . . . . 10 |
30 | xaddpnf1 9629 | . . . . . . . . . 10 | |
31 | 27, 29, 30 | syl2anc 408 | . . . . . . . . 9 |
32 | 23, 31 | sylan9eqr 2194 | . . . . . . . 8 |
33 | xaddpnf1 9629 | . . . . . . . . . 10 | |
34 | 33 | 3ad2ant1 1002 | . . . . . . . . 9 |
35 | 34 | adantr 274 | . . . . . . . 8 |
36 | 32, 35 | eqtr4d 2175 | . . . . . . 7 |
37 | oveq2 5782 | . . . . . . . . 9 | |
38 | xaddpnf1 9629 | . . . . . . . . . 10 | |
39 | 38 | 3ad2ant2 1003 | . . . . . . . . 9 |
40 | 37, 39 | sylan9eqr 2194 | . . . . . . . 8 |
41 | 40 | oveq2d 5790 | . . . . . . 7 |
42 | 36, 41 | eqtr4d 2175 | . . . . . 6 |
43 | 42 | adantlr 468 | . . . . 5 |
44 | simp3 983 | . . . . . . 7 | |
45 | xrnemnf 9564 | . . . . . . 7 | |
46 | 44, 45 | sylib 121 | . . . . . 6 |
47 | 46 | adantr 274 | . . . . 5 |
48 | 22, 43, 47 | mpjaodan 787 | . . . 4 |
49 | 48 | anassrs 397 | . . 3 |
50 | xaddpnf2 9630 | . . . . . . . 8 | |
51 | 50 | 3ad2ant3 1004 | . . . . . . 7 |
52 | 51, 34 | eqtr4d 2175 | . . . . . 6 |
53 | 52 | adantr 274 | . . . . 5 |
54 | oveq2 5782 | . . . . . . 7 | |
55 | 54, 34 | sylan9eqr 2194 | . . . . . 6 |
56 | 55 | oveq1d 5789 | . . . . 5 |
57 | oveq1 5781 | . . . . . . 7 | |
58 | 57, 51 | sylan9eqr 2194 | . . . . . 6 |
59 | 58 | oveq2d 5790 | . . . . 5 |
60 | 53, 56, 59 | 3eqtr4d 2182 | . . . 4 |
61 | 60 | adantlr 468 | . . 3 |
62 | simpl2 985 | . . . 4 | |
63 | xrnemnf 9564 | . . . 4 | |
64 | 62, 63 | sylib 121 | . . 3 |
65 | 49, 61, 64 | mpjaodan 787 | . 2 |
66 | simpl3 986 | . . . . 5 | |
67 | 66, 50 | syl 14 | . . . 4 |
68 | simpl2l 1034 | . . . . . 6 | |
69 | simpl3l 1036 | . . . . . 6 | |
70 | xaddcl 9643 | . . . . . 6 | |
71 | 68, 69, 70 | syl2anc 408 | . . . . 5 |
72 | simpl2 985 | . . . . . 6 | |
73 | xaddnemnf 9640 | . . . . . 6 | |
74 | 72, 66, 73 | syl2anc 408 | . . . . 5 |
75 | xaddpnf2 9630 | . . . . 5 | |
76 | 71, 74, 75 | syl2anc 408 | . . . 4 |
77 | 67, 76 | eqtr4d 2175 | . . 3 |
78 | simpr 109 | . . . . . 6 | |
79 | 78 | oveq1d 5789 | . . . . 5 |
80 | xaddpnf2 9630 | . . . . . 6 | |
81 | 72, 80 | syl 14 | . . . . 5 |
82 | 79, 81 | eqtrd 2172 | . . . 4 |
83 | 82 | oveq1d 5789 | . . 3 |
84 | 78 | oveq1d 5789 | . . 3 |
85 | 77, 83, 84 | 3eqtr4d 2182 | . 2 |
86 | simp1 981 | . . 3 | |
87 | xrnemnf 9564 | . . 3 | |
88 | 86, 87 | sylib 121 | . 2 |
89 | 65, 85, 88 | mpjaodan 787 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 w3a 962 wceq 1331 wcel 1480 wne 2308 (class class class)co 5774 cc 7618 cr 7619 caddc 7623 cpnf 7797 cmnf 7798 cxr 7799 cxad 9557 |
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 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-addass 7722 ax-rnegex 7729 |
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 7802 df-mnf 7803 df-xr 7804 df-xadd 9560 |
This theorem is referenced by: xaddass2 9653 xpncan 9654 xadd4d 9668 |
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