Theorem xrpxdivcld 29643

Description: Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)

Hypotheses

Ref	Expression
xrpxdivcld.1	|- ( ph -> A e. ( 0 [,] +oo ) )
xrpxdivcld.2	|- ( ph -> B e. RR+ )

Assertion

Ref	Expression
xrpxdivcld	|- ( ph -> ( A /𝑒 B ) e. ( 0 [,] +oo ) )

Proof of Theorem xrpxdivcld

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( A = 0 -> ( A /𝑒 B ) = ( 0 /𝑒 B ) )
2		xrpxdivcld.2	. . . . 5 |- ( ph -> B e. RR+ )
3		xdiv0rp 29638	. . . . 5 |- ( B e. RR+ -> ( 0 /𝑒 B ) = 0 )
4	2, 3	syl 17	. . . 4 |- ( ph -> ( 0 /𝑒 B ) = 0 )
5	1, 4	sylan9eqr 2678	. . 3 |- ( ( ph /\ A = 0 ) -> ( A /𝑒 B ) = 0 )
6		elxrge02 29640	. . . . 5 |- ( ( A /𝑒 B ) e. ( 0 [,] +oo ) <-> ( ( A /𝑒 B ) = 0 \/ ( A /𝑒 B ) e. RR+ \/ ( A /𝑒 B ) = +oo ) )
7	6	biimpri 218	. . . 4 |- ( ( ( A /𝑒 B ) = 0 \/ ( A /𝑒 B ) e. RR+ \/ ( A /𝑒 B ) = +oo ) -> ( A /𝑒 B ) e. ( 0 [,] +oo ) )
8	7	3o1cs 29309	. . 3 |- ( ( A /𝑒 B ) = 0 -> ( A /𝑒 B ) e. ( 0 [,] +oo ) )
9	5, 8	syl 17	. 2 |- ( ( ph /\ A = 0 ) -> ( A /𝑒 B ) e. ( 0 [,] +oo ) )
10		simpr 477	. . . 4 |- ( ( ph /\ A e. RR+ ) -> A e. RR+ )
11	2	adantr 481	. . . 4 |- ( ( ph /\ A e. RR+ ) -> B e. RR+ )
12	10, 11	rpxdivcld 29642	. . 3 |- ( ( ph /\ A e. RR+ ) -> ( A /𝑒 B ) e. RR+ )
13	7	3o2cs 29310	. . 3 |- ( ( A /𝑒 B ) e. RR+ -> ( A /𝑒 B ) e. ( 0 [,] +oo ) )
14	12, 13	syl 17	. 2 |- ( ( ph /\ A e. RR+ ) -> ( A /𝑒 B ) e. ( 0 [,] +oo ) )
15		oveq1 6657	. . . 4 |- ( A = +oo -> ( A /𝑒 B ) = ( +oo /𝑒 B ) )
16		xdivpnfrp 29641	. . . . 5 |- ( B e. RR+ -> ( +oo /𝑒 B ) = +oo )
17	2, 16	syl 17	. . . 4 |- ( ph -> ( +oo /𝑒 B ) = +oo )
18	15, 17	sylan9eqr 2678	. . 3 |- ( ( ph /\ A = +oo ) -> ( A /𝑒 B ) = +oo )
19	7	3o3cs 29311	. . 3 |- ( ( A /𝑒 B ) = +oo -> ( A /𝑒 B ) e. ( 0 [,] +oo ) )
20	18, 19	syl 17	. 2 |- ( ( ph /\ A = +oo ) -> ( A /𝑒 B ) e. ( 0 [,] +oo ) )
21		xrpxdivcld.1	. . 3 |- ( ph -> A e. ( 0 [,] +oo ) )
22		elxrge02 29640	. . 3 |- ( A e. ( 0 [,] +oo ) <-> ( A = 0 \/ A e. RR+ \/ A = +oo ) )
23	21, 22	sylib 208	. 2 |- ( ph -> ( A = 0 \/ A e. RR+ \/ A = +oo ) )
24	9, 14, 20, 23	mpjao3dan 1395	1 |- ( ph -> ( A /𝑒 B ) e. ( 0 [,] +oo ) )

Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990  (class class class)co 6650   0cc0 9936   +oocpnf 10071   RR+crp 11832   [,]cicc 12178   /_𝑒 cxdiv 29625

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-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-xdiv 29626

This theorem is referenced by:  measdivcstOLD  30287  measdivcst  30288