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Mirrors > Home > ILE Home > Th. List > xaddass2 | Unicode version |
Description: Associativity of extended real addition. See xaddass 9796 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddass2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1010 | . . . . . 6 | |
2 | xnegcl 9759 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | simp1r 1011 | . . . . . . 7 | |
5 | pnfxr 7942 | . . . . . . . . 9 | |
6 | xneg11 9761 | . . . . . . . . 9 | |
7 | 1, 5, 6 | sylancl 410 | . . . . . . . 8 |
8 | 7 | necon3bid 2375 | . . . . . . 7 |
9 | 4, 8 | mpbird 166 | . . . . . 6 |
10 | xnegpnf 9755 | . . . . . . 7 | |
11 | 10 | a1i 9 | . . . . . 6 |
12 | 9, 11 | neeqtrd 2362 | . . . . 5 |
13 | simp2l 1012 | . . . . . 6 | |
14 | xnegcl 9759 | . . . . . 6 | |
15 | 13, 14 | syl 14 | . . . . 5 |
16 | simp2r 1013 | . . . . . . 7 | |
17 | xneg11 9761 | . . . . . . . . 9 | |
18 | 13, 5, 17 | sylancl 410 | . . . . . . . 8 |
19 | 18 | necon3bid 2375 | . . . . . . 7 |
20 | 16, 19 | mpbird 166 | . . . . . 6 |
21 | 20, 11 | neeqtrd 2362 | . . . . 5 |
22 | simp3l 1014 | . . . . . 6 | |
23 | xnegcl 9759 | . . . . . 6 | |
24 | 22, 23 | syl 14 | . . . . 5 |
25 | simp3r 1015 | . . . . . . 7 | |
26 | xneg11 9761 | . . . . . . . . 9 | |
27 | 22, 5, 26 | sylancl 410 | . . . . . . . 8 |
28 | 27 | necon3bid 2375 | . . . . . . 7 |
29 | 25, 28 | mpbird 166 | . . . . . 6 |
30 | 29, 11 | neeqtrd 2362 | . . . . 5 |
31 | xaddass 9796 | . . . . 5 | |
32 | 3, 12, 15, 21, 24, 30, 31 | syl222anc 1243 | . . . 4 |
33 | xnegdi 9795 | . . . . . 6 | |
34 | 1, 13, 33 | syl2anc 409 | . . . . 5 |
35 | 34 | oveq1d 5851 | . . . 4 |
36 | xnegdi 9795 | . . . . . 6 | |
37 | 13, 22, 36 | syl2anc 409 | . . . . 5 |
38 | 37 | oveq2d 5852 | . . . 4 |
39 | 32, 35, 38 | 3eqtr4d 2207 | . . 3 |
40 | xaddcl 9787 | . . . . 5 | |
41 | 1, 13, 40 | syl2anc 409 | . . . 4 |
42 | xnegdi 9795 | . . . 4 | |
43 | 41, 22, 42 | syl2anc 409 | . . 3 |
44 | xaddcl 9787 | . . . . 5 | |
45 | 13, 22, 44 | syl2anc 409 | . . . 4 |
46 | xnegdi 9795 | . . . 4 | |
47 | 1, 45, 46 | syl2anc 409 | . . 3 |
48 | 39, 43, 47 | 3eqtr4d 2207 | . 2 |
49 | xaddcl 9787 | . . . 4 | |
50 | 41, 22, 49 | syl2anc 409 | . . 3 |
51 | xaddcl 9787 | . . . 4 | |
52 | 1, 45, 51 | syl2anc 409 | . . 3 |
53 | xneg11 9761 | . . 3 | |
54 | 50, 52, 53 | syl2anc 409 | . 2 |
55 | 48, 54 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wne 2334 (class class class)co 5836 cpnf 7921 cmnf 7922 cxr 7923 cxne 9696 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-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
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-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 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-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-sub 8062 df-neg 8063 df-xneg 9699 df-xadd 9700 |
This theorem is referenced by: (None) |
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