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Mirrors > Home > ILE Home > Th. List > xaddass2 | Unicode version |
Description: Associativity of extended real addition. See xaddass 9652 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddass2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1005 | . . . . . 6 | |
2 | xnegcl 9615 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | simp1r 1006 | . . . . . . 7 | |
5 | pnfxr 7818 | . . . . . . . . 9 | |
6 | xneg11 9617 | . . . . . . . . 9 | |
7 | 1, 5, 6 | sylancl 409 | . . . . . . . 8 |
8 | 7 | necon3bid 2349 | . . . . . . 7 |
9 | 4, 8 | mpbird 166 | . . . . . 6 |
10 | xnegpnf 9611 | . . . . . . 7 | |
11 | 10 | a1i 9 | . . . . . 6 |
12 | 9, 11 | neeqtrd 2336 | . . . . 5 |
13 | simp2l 1007 | . . . . . 6 | |
14 | xnegcl 9615 | . . . . . 6 | |
15 | 13, 14 | syl 14 | . . . . 5 |
16 | simp2r 1008 | . . . . . . 7 | |
17 | xneg11 9617 | . . . . . . . . 9 | |
18 | 13, 5, 17 | sylancl 409 | . . . . . . . 8 |
19 | 18 | necon3bid 2349 | . . . . . . 7 |
20 | 16, 19 | mpbird 166 | . . . . . 6 |
21 | 20, 11 | neeqtrd 2336 | . . . . 5 |
22 | simp3l 1009 | . . . . . 6 | |
23 | xnegcl 9615 | . . . . . 6 | |
24 | 22, 23 | syl 14 | . . . . 5 |
25 | simp3r 1010 | . . . . . . 7 | |
26 | xneg11 9617 | . . . . . . . . 9 | |
27 | 22, 5, 26 | sylancl 409 | . . . . . . . 8 |
28 | 27 | necon3bid 2349 | . . . . . . 7 |
29 | 25, 28 | mpbird 166 | . . . . . 6 |
30 | 29, 11 | neeqtrd 2336 | . . . . 5 |
31 | xaddass 9652 | . . . . 5 | |
32 | 3, 12, 15, 21, 24, 30, 31 | syl222anc 1232 | . . . 4 |
33 | xnegdi 9651 | . . . . . 6 | |
34 | 1, 13, 33 | syl2anc 408 | . . . . 5 |
35 | 34 | oveq1d 5789 | . . . 4 |
36 | xnegdi 9651 | . . . . . 6 | |
37 | 13, 22, 36 | syl2anc 408 | . . . . 5 |
38 | 37 | oveq2d 5790 | . . . 4 |
39 | 32, 35, 38 | 3eqtr4d 2182 | . . 3 |
40 | xaddcl 9643 | . . . . 5 | |
41 | 1, 13, 40 | syl2anc 408 | . . . 4 |
42 | xnegdi 9651 | . . . 4 | |
43 | 41, 22, 42 | syl2anc 408 | . . 3 |
44 | xaddcl 9643 | . . . . 5 | |
45 | 13, 22, 44 | syl2anc 408 | . . . 4 |
46 | xnegdi 9651 | . . . 4 | |
47 | 1, 45, 46 | syl2anc 408 | . . 3 |
48 | 39, 43, 47 | 3eqtr4d 2182 | . 2 |
49 | xaddcl 9643 | . . . 4 | |
50 | 41, 22, 49 | syl2anc 408 | . . 3 |
51 | xaddcl 9643 | . . . 4 | |
52 | 1, 45, 51 | syl2anc 408 | . . 3 |
53 | xneg11 9617 | . . 3 | |
54 | 50, 52, 53 | syl2anc 408 | . 2 |
55 | 48, 54 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wne 2308 (class class class)co 5774 cpnf 7797 cmnf 7798 cxr 7799 cxne 9556 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-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
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-reu 2423 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-riota 5730 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-sub 7935 df-neg 7936 df-xneg 9559 df-xadd 9560 |
This theorem is referenced by: (None) |
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