Theorem muc0or 6253

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.

Step	Hyp	Ref	Expression
1		elncs 6120	. . . . 5 ⊢ (A ∈ NC ↔ ∃x A = Nc x)
2		elncs 6120	. . . . 5 ⊢ (B ∈ NC ↔ ∃y B = Nc y)
3	1, 2	anbi12i 678	. . . 4 ⊢ ((A ∈ NC ∧ B ∈ NC ) ↔ (∃x A = Nc x ∧ ∃y B = Nc y))
4		eeanv 1913	. . . 4 ⊢ (∃x∃y(A = Nc x ∧ B = Nc y) ↔ (∃x A = Nc x ∧ ∃y B = Nc y))
5	3, 4	bitr4i 243	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ) ↔ ∃x∃y(A = Nc x ∧ B = Nc y))
6		vex 2863	. . . . . . . 8 ⊢ x ∈ V
7		vex 2863	. . . . . . . 8 ⊢ y ∈ V
8	6, 7	mucnc 6132	. . . . . . 7 ⊢ ( Nc x ·c Nc y) = Nc (x × y)
9		df0c2 6138	. . . . . . 7 ⊢ 0c = Nc ∅
10	8, 9	eqeq12i 2366	. . . . . 6 ⊢ (( Nc x ·c Nc y) = 0c ↔ Nc (x × y) = Nc ∅)
11	6, 7	xpex 5116	. . . . . . . . 9 ⊢ (x × y) ∈ V
12	11	eqnc 6128	. . . . . . . 8 ⊢ ( Nc (x × y) = Nc ∅ ↔ (x × y) ≈ ∅)
13		en0 6043	. . . . . . . 8 ⊢ ((x × y) ≈ ∅ ↔ (x × y) = ∅)
14	12, 13	bitri 240	. . . . . . 7 ⊢ ( Nc (x × y) = Nc ∅ ↔ (x × y) = ∅)
15		xpeq0 5047	. . . . . . . 8 ⊢ ((x × y) = ∅ ↔ (x = ∅ ∨ y = ∅))
16		nceq 6109	. . . . . . . . 9 ⊢ (x = ∅ → Nc x = Nc ∅)
17		nceq 6109	. . . . . . . . 9 ⊢ (y = ∅ → Nc y = Nc ∅)
18	16, 17	orim12i 502	. . . . . . . 8 ⊢ ((x = ∅ ∨ y = ∅) → ( Nc x = Nc ∅ ∨ Nc y = Nc ∅))
19	15, 18	sylbi 187	. . . . . . 7 ⊢ ((x × y) = ∅ → ( Nc x = Nc ∅ ∨ Nc y = Nc ∅))
20	14, 19	sylbi 187	. . . . . 6 ⊢ ( Nc (x × y) = Nc ∅ → ( Nc x = Nc ∅ ∨ Nc y = Nc ∅))
21	10, 20	sylbi 187	. . . . 5 ⊢ (( Nc x ·c Nc y) = 0c → ( Nc x = Nc ∅ ∨ Nc y = Nc ∅))
22		oveq12 5533	. . . . . . 7 ⊢ ((A = Nc x ∧ B = Nc y) → (A ·c B) = ( Nc x ·c Nc y))
23	22	eqeq1d 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 = 0c ↔ Nc x = 0c))
25	24	adantr 451	. . . . . . . 8 ⊢ ((A = Nc x ∧ B = Nc y) → (A = 0c ↔ Nc x = 0c))
26	9	eqeq2i 2363	. . . . . . . 8 ⊢ ( Nc x = 0c ↔ Nc x = Nc ∅)
27	25, 26	syl6bb 252	. . . . . . 7 ⊢ ((A = Nc x ∧ B = Nc y) → (A = 0c ↔ Nc x = Nc ∅))
28		eqeq1 2359	. . . . . . . . 9 ⊢ (B = Nc y → (B = 0c ↔ Nc y = 0c))
29	9	eqeq2i 2363	. . . . . . . . 9 ⊢ ( Nc y = 0c ↔ Nc y = Nc ∅)
30	28, 29	syl6bb 252	. . . . . . . 8 ⊢ (B = Nc y → (B = 0c ↔ Nc y = Nc ∅))
31	30	adantl 452	. . . . . . 7 ⊢ ((A = Nc x ∧ B = Nc y) → (B = 0c ↔ Nc y = Nc ∅))
32	27, 31	orbi12d 690	. . . . . 6 ⊢ ((A = Nc x ∧ B = Nc y) → ((A = 0c ∨ B = 0c) ↔ ( Nc x = Nc ∅ ∨ Nc y = Nc ∅)))
33	23, 32	imbi12d 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 ∅))))
34	21, 33	mpbiri 224	. . . 4 ⊢ ((A = Nc x ∧ B = Nc y) → ((A ·c B) = 0c → (A = 0c ∨ B = 0c)))
35	34	exlimivv 1635	. . 3 ⊢ (∃x∃y(A = Nc x ∧ B = Nc y) → ((A ·c B) = 0c → (A = 0c ∨ B = 0c)))
36	5, 35	sylbi 187	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A ·c B) = 0c → (A = 0c ∨ B = 0c)))
37		0cnc 6139	. . . . . . 7 ⊢ 0c ∈ NC
38		muccom 6135	. . . . . . 7 ⊢ ((0c ∈ NC ∧ B ∈ NC ) → (0c ·c B) = (B ·c 0c))
39	37, 38	mpan 651	. . . . . 6 ⊢ (B ∈ NC → (0c ·c B) = (B ·c 0c))
40		muc0 6143	. . . . . 6 ⊢ (B ∈ NC → (B ·c 0c) = 0c)
41	39, 40	eqtrd 2385	. . . . 5 ⊢ (B ∈ NC → (0c ·c B) = 0c)
42		oveq1 5531	. . . . . 6 ⊢ (A = 0c → (A ·c B) = (0c ·c B))
43	42	eqeq1d 2361	. . . . 5 ⊢ (A = 0c → ((A ·c B) = 0c ↔ (0c ·c B) = 0c))
44	41, 43	syl5ibrcom 213	. . . 4 ⊢ (B ∈ NC → (A = 0c → (A ·c B) = 0c))
45	44	adantl 452	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (A = 0c → (A ·c B) = 0c))
46		muc0 6143	. . . . 5 ⊢ (A ∈ NC → (A ·c 0c) = 0c)
47		oveq2 5532	. . . . . 6 ⊢ (B = 0c → (A ·c B) = (A ·c 0c))
48	47	eqeq1d 2361	. . . . 5 ⊢ (B = 0c → ((A ·c B) = 0c ↔ (A ·c 0c) = 0c))
49	46, 48	syl5ibrcom 213	. . . 4 ⊢ (A ∈ NC → (B = 0c → (A ·c B) = 0c))
50	49	adantr 451	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (B = 0c → (A ·c B) = 0c))
51	45, 50	jaod 369	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A = 0c ∨ B = 0c) → (A ·c B) = 0c))
52	36, 51	impbid 183	1 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A ·c B) = 0c ↔ (A = 0c ∨ B = 0c)))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∅c0 3551  0_cc0c 4375   class class class wbr 4640   × cxp 4771  (class class class)co 5526   ≈ cen 6029   NC cncs 6089   Nc cnc 6092   ·_c cmuc 6093

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-pprod 5739  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-cross 5765  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102  df-muc 6103

This theorem is referenced by: (None)