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Mirrors > Home > MPE Home > Th. List > xaddf | Unicode version |
Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xaddf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 9661 | . . . . . 6 | |
2 | pnfxr 11350 | . . . . . 6 | |
3 | 1, 2 | keepel 4009 | . . . . 5 |
4 | 3 | a1i 11 | . . . 4 |
5 | mnfxr 11352 | . . . . . . 7 | |
6 | 1, 5 | keepel 4009 | . . . . . 6 |
7 | 6 | a1i 11 | . . . . 5 |
8 | 2 | a1i 11 | . . . . . . . 8 |
9 | 5 | a1i 11 | . . . . . . . . 9 |
10 | ioran 490 | . . . . . . . . . . . . . 14 | |
11 | elxr 11354 | . . . . . . . . . . . . . . . . . 18 | |
12 | 3orass 976 | . . . . . . . . . . . . . . . . . 18 | |
13 | 11, 12 | sylbb 197 | . . . . . . . . . . . . . . . . 17 |
14 | 13 | ord 377 | . . . . . . . . . . . . . . . 16 |
15 | 14 | con1d 124 | . . . . . . . . . . . . . . 15 |
16 | 15 | imp 429 | . . . . . . . . . . . . . 14 |
17 | 10, 16 | sylan2br 476 | . . . . . . . . . . . . 13 |
18 | ioran 490 | . . . . . . . . . . . . . 14 | |
19 | elxr 11354 | . . . . . . . . . . . . . . . . . 18 | |
20 | 3orass 976 | . . . . . . . . . . . . . . . . . 18 | |
21 | 19, 20 | sylbb 197 | . . . . . . . . . . . . . . . . 17 |
22 | 21 | ord 377 | . . . . . . . . . . . . . . . 16 |
23 | 22 | con1d 124 | . . . . . . . . . . . . . . 15 |
24 | 23 | imp 429 | . . . . . . . . . . . . . 14 |
25 | 18, 24 | sylan2br 476 | . . . . . . . . . . . . 13 |
26 | readdcl 9596 | . . . . . . . . . . . . 13 | |
27 | 17, 25, 26 | syl2an 477 | . . . . . . . . . . . 12 |
28 | 27 | rexrd 9664 | . . . . . . . . . . 11 |
29 | 28 | anassrs 648 | . . . . . . . . . 10 |
30 | 29 | anassrs 648 | . . . . . . . . 9 |
31 | 9, 30 | ifclda 3973 | . . . . . . . 8 |
32 | 8, 31 | ifclda 3973 | . . . . . . 7 |
33 | 32 | an32s 804 | . . . . . 6 |
34 | 33 | anassrs 648 | . . . . 5 |
35 | 7, 34 | ifclda 3973 | . . . 4 |
36 | 4, 35 | ifclda 3973 | . . 3 |
37 | 36 | rgen2a 2884 | . 2 |
38 | df-xadd 11348 | . . 3 | |
39 | 38 | fmpt2 6867 | . 2 |
40 | 37, 39 | mpbi 208 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -. wn 3 \/ wo 368
/\ wa 369 \/ w3o 972 = wceq 1395
e. wcel 1818 A. wral 2807 if cif 3941
X. cxp 5002 --> wf 5589 (class class class)co 6296
cr 9512 0 cc0 9513 caddc 9516 cpnf 9646 cmnf 9647
cxr 9648
cxad 11345 |
This theorem is referenced by: xaddcl 11465 xrsadd 18435 xrofsup 27582 xrge0pluscn 27922 xrge0tmdOLD 27927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-cnex 9569 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-i2m1 9581 ax-1ne0 9582 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-id 4800 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-fv 5601 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-1st 6800 df-2nd 6801 df-pnf 9651 df-mnf 9652 df-xr 9653 df-xadd 11348 |
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