xaddf - Metamath Proof Explorer

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Theorem xaddf 12055

Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)

**Assertion**
Ref	Expression
xaddf

Proof of Theorem xaddf Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0xr 10086	. . . . . 6
2		pnfxr 10092	. . . . . 6
3	1, 2	keepel 4155	. . . . 5
4	3	a1i 11	. . . 4 $/\$ $/\$
5		mnfxr 10096	. . . . . . 7
6	1, 5	keepel 4155	. . . . . 6
7	6	a1i 11	. . . . 5 $/\$ $/\$ $/\$
8	2	a1i 11	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
9	5	a1i 11	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
10		ioran 511	. . . . . . . . . . . . . 14 $\/$ $/\$
11		elxr 11950	. . . . . . . . . . . . . . . . . 18 $\/$ $\/$
12		3orass 1040	. . . . . . . . . . . . . . . . . 18 $\/$ $\/$ $\/$ $\/$
13	11, 12	sylbb 209	. . . . . . . . . . . . . . . . 17 $\/$ $\/$
14	13	ord 392	. . . . . . . . . . . . . . . 16 $\/$
15	14	con1d 139	. . . . . . . . . . . . . . 15 $\/$
16	15	imp 445	. . . . . . . . . . . . . 14 $/\$ $\/$
17	10, 16	sylan2br 493	. . . . . . . . . . . . 13 $/\$ $/\$
18		ioran 511	. . . . . . . . . . . . . 14 $\/$ $/\$
19		elxr 11950	. . . . . . . . . . . . . . . . . 18 $\/$ $\/$
20		3orass 1040	. . . . . . . . . . . . . . . . . 18 $\/$ $\/$ $\/$ $\/$
21	19, 20	sylbb 209	. . . . . . . . . . . . . . . . 17 $\/$ $\/$
22	21	ord 392	. . . . . . . . . . . . . . . 16 $\/$
23	22	con1d 139	. . . . . . . . . . . . . . 15 $\/$
24	23	imp 445	. . . . . . . . . . . . . 14 $/\$ $\/$
25	18, 24	sylan2br 493	. . . . . . . . . . . . 13 $/\$ $/\$
26		readdcl 10019	. . . . . . . . . . . . 13 $/\$
27	17, 25, 26	syl2an 494	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
28	27	rexrd 10089	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
29	28	anassrs 680	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
30	29	anassrs 680	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
31	9, 30	ifclda 4120	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
32	8, 31	ifclda 4120	. . . . . . 7 $/\$ $/\$ $/\$
33	32	an32s 846	. . . . . 6 $/\$ $/\$ $/\$
34	33	anassrs 680	. . . . 5 $/\$ $/\$ $/\$
35	7, 34	ifclda 4120	. . . 4 $/\$ $/\$
36	4, 35	ifclda 4120	. . 3 $/\$
37	36	rgen2a 2977	. 2
38		df-xadd 11947	. . 3
39	38	fmpt2 7237	. 2
40	37, 39	mpbi 220	1

Colors of variables: wff setvar class

Syntax hints:

wn 3 $\/$ wo 383 $/\$ wa 384 $\/$ w3o 1036

wceq 1483

wcel 1990

wral 2912

cif 4086

cxp 5112

wf 5884 (class class class)co 6650

cr 9935

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-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009

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-ral 2917 df-rex 2918 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-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-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-pnf 10076 df-mnf 10077 df-xr 10078 df-xadd 11947

This theorem is referenced by: xaddcl 12070 xrsadd 19763 xrofsup 29533 xrge0pluscn 29986 xrge0tmdOLD 29991

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