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Mirrors > Home > MPE Home > Th. List > xaddass2 | Structured version Visualization version Unicode version |
Description: Associativity of extended real addition. See xaddass 12079 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddass2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1085 | . . . . . 6 | |
2 | xnegcl 12044 | . . . . . 6 | |
3 | 1, 2 | syl 17 | . . . . 5 |
4 | simp1r 1086 | . . . . . . 7 | |
5 | pnfxr 10092 | . . . . . . . . 9 | |
6 | xneg11 12046 | . . . . . . . . 9 | |
7 | 1, 5, 6 | sylancl 694 | . . . . . . . 8 |
8 | 7 | necon3bid 2838 | . . . . . . 7 |
9 | 4, 8 | mpbird 247 | . . . . . 6 |
10 | xnegpnf 12040 | . . . . . . 7 | |
11 | 10 | a1i 11 | . . . . . 6 |
12 | 9, 11 | neeqtrd 2863 | . . . . 5 |
13 | simp2l 1087 | . . . . . 6 | |
14 | xnegcl 12044 | . . . . . 6 | |
15 | 13, 14 | syl 17 | . . . . 5 |
16 | simp2r 1088 | . . . . . . 7 | |
17 | xneg11 12046 | . . . . . . . . 9 | |
18 | 13, 5, 17 | sylancl 694 | . . . . . . . 8 |
19 | 18 | necon3bid 2838 | . . . . . . 7 |
20 | 16, 19 | mpbird 247 | . . . . . 6 |
21 | 20, 11 | neeqtrd 2863 | . . . . 5 |
22 | simp3l 1089 | . . . . . 6 | |
23 | xnegcl 12044 | . . . . . 6 | |
24 | 22, 23 | syl 17 | . . . . 5 |
25 | simp3r 1090 | . . . . . . 7 | |
26 | xneg11 12046 | . . . . . . . . 9 | |
27 | 22, 5, 26 | sylancl 694 | . . . . . . . 8 |
28 | 27 | necon3bid 2838 | . . . . . . 7 |
29 | 25, 28 | mpbird 247 | . . . . . 6 |
30 | 29, 11 | neeqtrd 2863 | . . . . 5 |
31 | xaddass 12079 | . . . . 5 | |
32 | 3, 12, 15, 21, 24, 30, 31 | syl222anc 1342 | . . . 4 |
33 | xnegdi 12078 | . . . . . 6 | |
34 | 1, 13, 33 | syl2anc 693 | . . . . 5 |
35 | 34 | oveq1d 6665 | . . . 4 |
36 | xnegdi 12078 | . . . . . 6 | |
37 | 13, 22, 36 | syl2anc 693 | . . . . 5 |
38 | 37 | oveq2d 6666 | . . . 4 |
39 | 32, 35, 38 | 3eqtr4d 2666 | . . 3 |
40 | xaddcl 12070 | . . . . 5 | |
41 | 1, 13, 40 | syl2anc 693 | . . . 4 |
42 | xnegdi 12078 | . . . 4 | |
43 | 41, 22, 42 | syl2anc 693 | . . 3 |
44 | xaddcl 12070 | . . . . 5 | |
45 | 13, 22, 44 | syl2anc 693 | . . . 4 |
46 | xnegdi 12078 | . . . 4 | |
47 | 1, 45, 46 | syl2anc 693 | . . 3 |
48 | 39, 43, 47 | 3eqtr4d 2666 | . 2 |
49 | xaddcl 12070 | . . . 4 | |
50 | 41, 22, 49 | syl2anc 693 | . . 3 |
51 | xaddcl 12070 | . . . 4 | |
52 | 1, 45, 51 | syl2anc 693 | . . 3 |
53 | xneg11 12046 | . . 3 | |
54 | 50, 52, 53 | syl2anc 693 | . 2 |
55 | 48, 54 | mpbid 222 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 (class class class)co 6650 cpnf 10071 cmnf 10072 cxr 10073 cxne 11943 cxad 11944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-sub 10268 df-neg 10269 df-xneg 11946 df-xadd 11947 |
This theorem is referenced by: infleinflem1 39586 |
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