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

Theorem hashinfxadd 12561
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 12522 . . . . 5  |-  ( A  e.  V  ->  (
( # `  A )  e.  NN0  \/  ( # `
 A )  = +oo ) )
2 df-nel 2628 . . . . . . . . 9  |-  ( (
# `  A )  e/  NN0  <->  -.  ( # `  A
)  e.  NN0 )
32anbi2i 698 . . . . . . . 8  |-  ( ( ( ( # `  A
)  = +oo  \/  ( # `  A )  e.  NN0 )  /\  ( # `  A )  e/  NN0 )  <->  ( (
( # `  A )  = +oo  \/  ( # `
 A )  e. 
NN0 )  /\  -.  ( # `  A )  e.  NN0 ) )
4 pm5.61 717 . . . . . . . 8  |-  ( ( ( ( # `  A
)  = +oo  \/  ( # `  A )  e.  NN0 )  /\  -.  ( # `  A
)  e.  NN0 )  <->  ( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )
)
53, 4sylbb 200 . . . . . . 7  |-  ( ( ( ( # `  A
)  = +oo  \/  ( # `  A )  e.  NN0 )  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )
)
65ex 435 . . . . . 6  |-  ( ( ( # `  A
)  = +oo  \/  ( # `  A )  e.  NN0 )  -> 
( ( # `  A
)  e/  NN0  ->  (
( # `  A )  = +oo  /\  -.  ( # `  A )  e.  NN0 ) ) )
76orcoms 390 . . . . 5  |-  ( ( ( # `  A
)  e.  NN0  \/  ( # `  A )  = +oo )  -> 
( ( # `  A
)  e/  NN0  ->  (
( # `  A )  = +oo  /\  -.  ( # `  A )  e.  NN0 ) ) )
81, 7syl 17 . . . 4  |-  ( A  e.  V  ->  (
( # `  A )  e/  NN0  ->  ( (
# `  A )  = +oo  /\  -.  ( # `
 A )  e. 
NN0 ) ) )
98imp 430 . . 3  |-  ( ( A  e.  V  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )
)
1093adant2 1024 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )
)
11 oveq1 6312 . . . . 5  |-  ( (
# `  A )  = +oo  ->  ( ( # `
 A ) +e ( # `  B
) )  =  ( +oo +e (
# `  B )
) )
12 hashxrcl 12536 . . . . . . . 8  |-  ( B  e.  W  ->  ( # `
 B )  e. 
RR* )
13 hashnemnf 12524 . . . . . . . 8  |-  ( B  e.  W  ->  ( # `
 B )  =/= -oo )
1412, 13jca 534 . . . . . . 7  |-  ( B  e.  W  ->  (
( # `  B )  e.  RR*  /\  ( # `
 B )  =/= -oo ) )
15143ad2ant2 1027 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  B
)  e.  RR*  /\  ( # `
 B )  =/= -oo ) )
16 xaddpnf2 11520 . . . . . 6  |-  ( ( ( # `  B
)  e.  RR*  /\  ( # `
 B )  =/= -oo )  ->  ( +oo +e ( # `  B ) )  = +oo )
1715, 16syl 17 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( +oo +e (
# `  B )
)  = +oo )
1811, 17sylan9eqr 2492 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A
)  e/  NN0 )  /\  ( # `  A )  = +oo )  -> 
( ( # `  A
) +e (
# `  B )
)  = +oo )
1918expcom 436 . . 3  |-  ( (
# `  A )  = +oo  ->  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
) +e (
# `  B )
)  = +oo )
)
2019adantr 466 . 2  |-  ( ( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )  ->  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  ->  ( ( # `  A ) +e
( # `  B ) )  = +oo )
)
2110, 20mpcom 37 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 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625    e/ wnel 2626   ` cfv 5601  (class class class)co 6305   +oocpnf 9671   -oocmnf 9672   RR*cxr 9673   NN0cn0 10869   +ecxad 11407   #chash 12512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-xadd 11410  df-hash 12513
This theorem is referenced by:  hashunx  12562  vdgrf  25471
  Copyright terms: Public domain W3C validator