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| Mirrors > Home > MPE Home > Th. List > mnfaddpnf | Structured version Visualization version Unicode version | ||
| Description: Addition of negative and positive infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| Ref | Expression |
|---|---|
| mnfaddpnf |
    
    |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mnfxr 10096 |
. . 3
  | |
| 2 | pnfxr 10092 |
. . 3
| |
| 3 | xaddval 12054 |
. . 3
![/\](wedge.gif "/\"){kind=small}   
| |
| 4 | 1, 2, 3 | mp2an 708 |
. 2
|
| 5 | mnfnepnf 10095 |
. . . 4
 | |
| 6 | ifnefalse 4098 |
. . . 4
| |
| 7 | 5, 6 | ax-mp 5 |
. . 3
|
| 8 | eqid 2622 |
. . . . 5
| |
| 9 | 8 | iftruei 4093 |
. . . 4
|
| 10 | eqid 2622 |
. . . . 5
| |
| 11 | 10 | iftruei 4093 |
. . . 4
|
| 12 | 9, 11 | eqtri 2644 |
. . 3
|
| 13 | 7, 12 | eqtri 2644 |
. 2
|
| 14 | 4, 13 | eqtri 2644 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wceq 1483 wcel 1990 wne 2794 cif 4086 (class class class)co 6650
cc0 9936
caddc 9939 cpnf 10071 cmnf 10072 cxr 10073 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-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-i2m1 10004 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-rab 2921 df-v 3202 df-sbc 3436 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-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pnf 10076 df-mnf 10077 df-xr 10078 df-xadd 11947 |
| This theorem is referenced by: xnegid 12069 xaddcom 12071 xnegdi 12078 xsubge0 12091 xadddilem 12124 xrsnsgrp 19782 |
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