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Theorem hashinfxadd 12433
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 12395 . . . . 5  |-  ( A  e.  V  ->  (
( # `  A )  e.  NN0  \/  ( # `
 A )  = +oo ) )
2 df-nel 2665 . . . . . . . . 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 1015 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  A
)  = +oo  /\  -.  ( # `  A
)  e.  NN0 )
)
11 oveq1 6302 . . . . 5  |-  ( (
# `  A )  = +oo  ->  ( ( # `
 A ) +e ( # `  B
) )  =  ( +oo +e (
# `  B )
) )
12 hashxrcl 12409 . . . . . . . 8  |-  ( B  e.  W  ->  ( # `
 B )  e. 
RR* )
13 hashnemnf 12397 . . . . . . . 8  |-  ( B  e.  W  ->  ( # `
 B )  =/= -oo )
1412, 13jca 532 . . . . . . 7  |-  ( B  e.  W  ->  (
( # `  B )  e.  RR*  /\  ( # `
 B )  =/= -oo ) )
15143ad2ant2 1018 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  -> 
( ( # `  B
)  e.  RR*  /\  ( # `
 B )  =/= -oo ) )
16 xaddpnf2 11438 . . . . . 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 2530 . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662    e/ wnel 2663   ` cfv 5594  (class class class)co 6295   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639   NN0cn0 10807   +ecxad 11328   #chash 12385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-xadd 11331  df-hash 12386
This theorem is referenced by:  hashunx  12434  vdgrf  24712
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