Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > xaddval | Structured version Visualization version Unicode version |
Description: Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . . 4 | |
2 | 1 | eqeq1d 2624 | . . 3 |
3 | simpr 477 | . . . . 5 | |
4 | 3 | eqeq1d 2624 | . . . 4 |
5 | 4 | ifbid 4108 | . . 3 |
6 | 1 | eqeq1d 2624 | . . . 4 |
7 | 3 | eqeq1d 2624 | . . . . 5 |
8 | 7 | ifbid 4108 | . . . 4 |
9 | oveq12 6659 | . . . . . 6 | |
10 | 4, 9 | ifbieq2d 4111 | . . . . 5 |
11 | 7, 10 | ifbieq2d 4111 | . . . 4 |
12 | 6, 8, 11 | ifbieq12d 4113 | . . 3 |
13 | 2, 5, 12 | ifbieq12d 4113 | . 2 |
14 | df-xadd 11947 | . 2 | |
15 | c0ex 10034 | . . . 4 | |
16 | pnfex 10093 | . . . 4 | |
17 | 15, 16 | ifex 4156 | . . 3 |
18 | mnfxr 10096 | . . . . . 6 | |
19 | 18 | elexi 3213 | . . . . 5 |
20 | 15, 19 | ifex 4156 | . . . 4 |
21 | ovex 6678 | . . . . . 6 | |
22 | 19, 21 | ifex 4156 | . . . . 5 |
23 | 16, 22 | ifex 4156 | . . . 4 |
24 | 20, 23 | ifex 4156 | . . 3 |
25 | 17, 24 | ifex 4156 | . 2 |
26 | 13, 14, 25 | ovmpt2a 6791 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 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-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: xaddpnf1 12057 xaddpnf2 12058 xaddmnf1 12059 xaddmnf2 12060 pnfaddmnf 12061 mnfaddpnf 12062 rexadd 12063 |
Copyright terms: Public domain | W3C validator |