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Mirrors > Home > MPE Home > Th. List > xaddpnf1 | Structured version Visualization version Unicode version |
Description: Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddpnf1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 10092 | . . 3 | |
2 | xaddval 12054 | . . 3 | |
3 | 1, 2 | mpan2 707 | . 2 |
4 | pnfnemnf 10094 | . . . . 5 | |
5 | ifnefalse 4098 | . . . . 5 | |
6 | 4, 5 | mp1i 13 | . . . 4 |
7 | ifnefalse 4098 | . . . . 5 | |
8 | eqid 2622 | . . . . . 6 | |
9 | 8 | iftruei 4093 | . . . . 5 |
10 | 7, 9 | syl6eq 2672 | . . . 4 |
11 | 6, 10 | ifeq12d 4106 | . . 3 |
12 | ifid 4125 | . . 3 | |
13 | 11, 12 | syl6eq 2672 | . 2 |
14 | 3, 13 | sylan9eq 2676 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 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: xnn0xaddcl 12066 xaddnemnf 12067 xaddcom 12071 xnn0xadd0 12077 xnegdi 12078 xaddass 12079 xleadd1a 12083 xlt2add 12090 xsubge0 12091 xlesubadd 12093 xadddilem 12124 xrsdsreclblem 19792 isxmet2d 22132 xrge0iifhom 29983 esumpr2 30129 hasheuni 30147 carsgclctunlem2 30381 ovolsplit 40205 sge0pr 40611 sge0split 40626 sge0xadd 40652 |
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