New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  muc0or GIF version

Theorem muc0or 6252
 Description: The cardinal product of two cardinal numbers is zero iff one of the numbers is zero. Biconditional form of theorem XI.2.34 of [Rosser] p. 380. (Contributed by Scott Fenton, 31-Jul-2019.)
Assertion
Ref Expression
muc0or ((A NC B NC ) → ((A ·c B) = 0c ↔ (A = 0c B = 0c)))

Proof of Theorem muc0or
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elncs 6119 . . . . 5 (A NCx A = Nc x)
2 elncs 6119 . . . . 5 (B NCy B = Nc y)
31, 2anbi12i 678 . . . 4 ((A NC B NC ) ↔ (x A = Nc x y B = Nc y))
4 eeanv 1913 . . . 4 (xy(A = Nc x B = Nc y) ↔ (x A = Nc x y B = Nc y))
53, 4bitr4i 243 . . 3 ((A NC B NC ) ↔ xy(A = Nc x B = Nc y))
6 vex 2862 . . . . . . . 8 x V
7 vex 2862 . . . . . . . 8 y V
86, 7mucnc 6131 . . . . . . 7 ( Nc x ·c Nc y) = Nc (x × y)
9 df0c2 6137 . . . . . . 7 0c = Nc
108, 9eqeq12i 2366 . . . . . 6 (( Nc x ·c Nc y) = 0cNc (x × y) = Nc )
116, 7xpex 5115 . . . . . . . . 9 (x × y) V
1211eqnc 6127 . . . . . . . 8 ( Nc (x × y) = Nc ↔ (x × y) ≈ )
13 en0 6042 . . . . . . . 8 ((x × y) ≈ ↔ (x × y) = )
1412, 13bitri 240 . . . . . . 7 ( Nc (x × y) = Nc ↔ (x × y) = )
15 xpeq0 5046 . . . . . . . 8 ((x × y) = ↔ (x = y = ))
16 nceq 6108 . . . . . . . . 9 (x = Nc x = Nc )
17 nceq 6108 . . . . . . . . 9 (y = Nc y = Nc )
1816, 17orim12i 502 . . . . . . . 8 ((x = y = ) → ( Nc x = Nc Nc y = Nc ))
1915, 18sylbi 187 . . . . . . 7 ((x × y) = → ( Nc x = Nc Nc y = Nc ))
2014, 19sylbi 187 . . . . . 6 ( Nc (x × y) = Nc → ( Nc x = Nc Nc y = Nc ))
2110, 20sylbi 187 . . . . 5 (( Nc x ·c Nc y) = 0c → ( Nc x = Nc Nc y = Nc ))
22 oveq12 5532 . . . . . . 7 ((A = Nc x B = Nc y) → (A ·c B) = ( Nc x ·c Nc y))
2322eqeq1d 2361 . . . . . 6 ((A = Nc x B = Nc y) → ((A ·c B) = 0c ↔ ( Nc x ·c Nc y) = 0c))
24 eqeq1 2359 . . . . . . . . 9 (A = Nc x → (A = 0cNc x = 0c))
2524adantr 451 . . . . . . . 8 ((A = Nc x B = Nc y) → (A = 0cNc x = 0c))
269eqeq2i 2363 . . . . . . . 8 ( Nc x = 0cNc x = Nc )
2725, 26syl6bb 252 . . . . . . 7 ((A = Nc x B = Nc y) → (A = 0cNc x = Nc ))
28 eqeq1 2359 . . . . . . . . 9 (B = Nc y → (B = 0cNc y = 0c))
299eqeq2i 2363 . . . . . . . . 9 ( Nc y = 0cNc y = Nc )
3028, 29syl6bb 252 . . . . . . . 8 (B = Nc y → (B = 0cNc y = Nc ))
3130adantl 452 . . . . . . 7 ((A = Nc x B = Nc y) → (B = 0cNc y = Nc ))
3227, 31orbi12d 690 . . . . . 6 ((A = Nc x B = Nc y) → ((A = 0c B = 0c) ↔ ( Nc x = Nc Nc y = Nc )))
3323, 32imbi12d 311 . . . . 5 ((A = Nc x B = Nc y) → (((A ·c B) = 0c → (A = 0c B = 0c)) ↔ (( Nc x ·c Nc y) = 0c → ( Nc x = Nc Nc y = Nc ))))
3421, 33mpbiri 224 . . . 4 ((A = Nc x B = Nc y) → ((A ·c B) = 0c → (A = 0c B = 0c)))
3534exlimivv 1635 . . 3 (xy(A = Nc x B = Nc y) → ((A ·c B) = 0c → (A = 0c B = 0c)))
365, 35sylbi 187 . 2 ((A NC B NC ) → ((A ·c B) = 0c → (A = 0c B = 0c)))
37 0cnc 6138 . . . . . . 7 0c NC
38 muccom 6134 . . . . . . 7 ((0c NC B NC ) → (0c ·c B) = (B ·c 0c))
3937, 38mpan 651 . . . . . 6 (B NC → (0c ·c B) = (B ·c 0c))
40 muc0 6142 . . . . . 6 (B NC → (B ·c 0c) = 0c)
4139, 40eqtrd 2385 . . . . 5 (B NC → (0c ·c B) = 0c)
42 oveq1 5530 . . . . . 6 (A = 0c → (A ·c B) = (0c ·c B))
4342eqeq1d 2361 . . . . 5 (A = 0c → ((A ·c B) = 0c ↔ (0c ·c B) = 0c))
4441, 43syl5ibrcom 213 . . . 4 (B NC → (A = 0c → (A ·c B) = 0c))
4544adantl 452 . . 3 ((A NC B NC ) → (A = 0c → (A ·c B) = 0c))
46 muc0 6142 . . . . 5 (A NC → (A ·c 0c) = 0c)
47 oveq2 5531 . . . . . 6 (B = 0c → (A ·c B) = (A ·c 0c))
4847eqeq1d 2361 . . . . 5 (B = 0c → ((A ·c B) = 0c ↔ (A ·c 0c) = 0c))
4946, 48syl5ibrcom 213 . . . 4 (A NC → (B = 0c → (A ·c B) = 0c))
5049adantr 451 . . 3 ((A NC B NC ) → (B = 0c → (A ·c B) = 0c))
5145, 50jaod 369 . 2 ((A NC B NC ) → ((A = 0c B = 0c) → (A ·c B) = 0c))
5236, 51impbid 183 1 ((A NC B NC ) → ((A ·c B) = 0c ↔ (A = 0c B = 0c)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∅c0 3550  0cc0c 4374   class class class wbr 4639   × cxp 4770  (class class class)co 5525   ≈ cen 6028   NC cncs 6088   Nc cnc 6091   ·c cmuc 6092 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-pprod 5738  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-cross 5764  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101  df-muc 6102 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator