MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hashinfxadd Structured version   Unicode version

Theorem hashinfxadd 12267
Description: The extended real addition of the size of an infinite set with the size of an arbitrary set yields plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
Assertion
Ref Expression
hashinfxadd  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
) +e (
# `  B )
)  = +oo )

Proof of Theorem hashinfxadd
StepHypRef Expression
1 hashnn0pnf 12231 . . . . 5  |-  ( A  e.  V  ->  (
( # `  A )  e.  NN0  \/  ( # `
 A )  = +oo ) )
2 df-nel 2651 . . . . . . . . 9  |-  ( (
# `  A )  e/  NN0  <->  -.  ( # `  A
)  e.  NN0 )
32anbi2i 694 . . . . . . . 8  |-  ( ( ( ( # `  A
)  = +oo  \/  ( # `  A )  e.  NN0 )  /\  ( # `  A )  e/  NN0 )  <->  ( (
( # `  A )  = +oo  \/  ( # `
 A )  e. 
NN0 )  /\  -.  ( # `  A )  e.  NN0 ) )
4 pm5.61 712 . . . . . . . 8  |-  ( ( ( ( # `  A
)  = +oo  \/  ( # `  A )  e.  NN0 )  /\  -.  ( # `  A
)  e.  NN0 )  <->  ( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )
)
53, 4sylbb 197 . . . . . . 7  |-  ( ( ( ( # `  A
)  = +oo  \/  ( # `  A )  e.  NN0 )  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )
)
65ex 434 . . . . . 6  |-  ( ( ( # `  A
)  = +oo  \/  ( # `  A )  e.  NN0 )  -> 
( ( # `  A
)  e/  NN0  ->  (
( # `  A )  = +oo  /\  -.  ( # `  A )  e.  NN0 ) ) )
76orcoms 389 . . . . 5  |-  ( ( ( # `  A
)  e.  NN0  \/  ( # `  A )  = +oo )  -> 
( ( # `  A
)  e/  NN0  ->  (
( # `  A )  = +oo  /\  -.  ( # `  A )  e.  NN0 ) ) )
81, 7syl 16 . . . 4  |-  ( A  e.  V  ->  (
( # `  A )  e/  NN0  ->  ( (
# `  A )  = +oo  /\  -.  ( # `
 A )  e. 
NN0 ) ) )
98imp 429 . . 3  |-  ( ( A  e.  V  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )
)
1093adant2 1007 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )
)
11 oveq1 6208 . . . . 5  |-  ( (
# `  A )  = +oo  ->  ( ( # `
 A ) +e ( # `  B
) )  =  ( +oo +e (
# `  B )
) )
12 hashxrcl 12245 . . . . . . . 8  |-  ( B  e.  W  ->  ( # `
 B )  e. 
RR* )
13 hashnemnf 12233 . . . . . . . 8  |-  ( B  e.  W  ->  ( # `
 B )  =/= -oo )
1412, 13jca 532 . . . . . . 7  |-  ( B  e.  W  ->  (
( # `  B )  e.  RR*  /\  ( # `
 B )  =/= -oo ) )
15143ad2ant2 1010 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  B
)  e.  RR*  /\  ( # `
 B )  =/= -oo ) )
16 xaddpnf2 11309 . . . . . 6  |-  ( ( ( # `  B
)  e.  RR*  /\  ( # `
 B )  =/= -oo )  ->  ( +oo +e ( # `  B ) )  = +oo )
1715, 16syl 16 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( +oo +e (
# `  B )
)  = +oo )
1811, 17sylan9eqr 2517 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A
)  e/  NN0 )  /\  ( # `  A )  = +oo )  -> 
( ( # `  A
) +e (
# `  B )
)  = +oo )
1918expcom 435 . . 3  |-  ( (
# `  A )  = +oo  ->  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
) +e (
# `  B )
)  = +oo )
)
2019adantr 465 . 2  |-  ( ( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )  ->  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  ->  ( ( # `  A ) +e
( # `  B ) )  = +oo )
)
2110, 20mpcom 36 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
) +e (
# `  B )
)  = +oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648    e/ wnel 2649   ` cfv 5527  (class class class)co 6201   +oocpnf 9527   -oocmnf 9528   RR*cxr 9529   NN0cn0 10691   +ecxad 11199   #chash 12221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-xadd 11202  df-hash 12222
This theorem is referenced by:  hashunx  12268  vdgrf  23721
  Copyright terms: Public domain W3C validator