Theorem ovmuc 6131

Description: The value of cardinal multiplication. (Contributed by SF, 10-Mar-2015.)

Assertion

Ref	Expression
ovmuc	⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ·c N) = {a ∣ ∃b ∈ M ∃g ∈ N a ≈ (b × g)})

Distinct variable groups:   a,b,g   M,a,b   N,a,b,g

Allowed substitution hint:   M(g)

Proof of Theorem ovmuc

Dummy variables c m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elima 4755	. . . . 5 ⊢ (a ∈ ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N) “ M) ↔ ∃b ∈ M b(ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N)a)
2		df-br 4641	. . . . . . 7 ⊢ (b(ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N)a ↔ ⟨b, a⟩ ∈ (ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N))
3		elima 4755	. . . . . . 7 ⊢ (⟨b, a⟩ ∈ (ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N) ↔ ∃g ∈ N gran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ )⟨b, a⟩)
4		df-br 4641	. . . . . . . . 9 ⊢ (gran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ )⟨b, a⟩ ↔ ⟨g, ⟨b, a⟩⟩ ∈ ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ))
5		elrn2 4898	. . . . . . . . . 10 ⊢ (⟨g, ⟨b, a⟩⟩ ∈ ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) ↔ ∃c⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ))
6		elin 3220	. . . . . . . . . . . 12 ⊢ (⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) ↔ (⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∧ ⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ Ins2 Ins2 ◡ ≈ ))
7		vex 2863	. . . . . . . . . . . . . . 15 ⊢ a ∈ V
8	7	oqelins4 5795	. . . . . . . . . . . . . 14 ⊢ (⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ↔ ⟨c, ⟨g, b⟩⟩ ∈ ran ( Cross ⊗ (2nd ⊗ 1st )))
9		elrn 4897	. . . . . . . . . . . . . . 15 ⊢ (⟨c, ⟨g, b⟩⟩ ∈ ran ( Cross ⊗ (2nd ⊗ 1st )) ↔ ∃a a( Cross ⊗ (2nd ⊗ 1st ))⟨c, ⟨g, b⟩⟩)
10		trtxp 5782	. . . . . . . . . . . . . . . . 17 ⊢ (a( Cross ⊗ (2nd ⊗ 1st ))⟨c, ⟨g, b⟩⟩ ↔ (a Cross c ∧ a(2nd ⊗ 1st )⟨g, b⟩))
11		trtxp 5782	. . . . . . . . . . . . . . . . . . 19 ⊢ (a(2nd ⊗ 1st )⟨g, b⟩ ↔ (a2nd g ∧ a1st b))
12		ancom 437	. . . . . . . . . . . . . . . . . . 19 ⊢ ((a2nd g ∧ a1st b) ↔ (a1st b ∧ a2nd g))
13		vex 2863	. . . . . . . . . . . . . . . . . . . 20 ⊢ b ∈ V
14		vex 2863	. . . . . . . . . . . . . . . . . . . 20 ⊢ g ∈ V
15	13, 14	op1st2nd 5791	. . . . . . . . . . . . . . . . . . 19 ⊢ ((a1st b ∧ a2nd g) ↔ a = ⟨b, g⟩)
16	11, 12, 15	3bitri 262	. . . . . . . . . . . . . . . . . 18 ⊢ (a(2nd ⊗ 1st )⟨g, b⟩ ↔ a = ⟨b, g⟩)
17	16	anbi2i 675	. . . . . . . . . . . . . . . . 17 ⊢ ((a Cross c ∧ a(2nd ⊗ 1st )⟨g, b⟩) ↔ (a Cross c ∧ a = ⟨b, g⟩))
18		ancom 437	. . . . . . . . . . . . . . . . 17 ⊢ ((a Cross c ∧ a = ⟨b, g⟩) ↔ (a = ⟨b, g⟩ ∧ a Cross c))
19	10, 17, 18	3bitri 262	. . . . . . . . . . . . . . . 16 ⊢ (a( Cross ⊗ (2nd ⊗ 1st ))⟨c, ⟨g, b⟩⟩ ↔ (a = ⟨b, g⟩ ∧ a Cross c))
20	19	exbii 1582	. . . . . . . . . . . . . . 15 ⊢ (∃a a( Cross ⊗ (2nd ⊗ 1st ))⟨c, ⟨g, b⟩⟩ ↔ ∃a(a = ⟨b, g⟩ ∧ a Cross c))
21	13, 14	opex 4589	. . . . . . . . . . . . . . . 16 ⊢ ⟨b, g⟩ ∈ V
22		breq1 4643	. . . . . . . . . . . . . . . 16 ⊢ (a = ⟨b, g⟩ → (a Cross c ↔ ⟨b, g⟩ Cross c))
23	21, 22	ceqsexv 2895	. . . . . . . . . . . . . . 15 ⊢ (∃a(a = ⟨b, g⟩ ∧ a Cross c) ↔ ⟨b, g⟩ Cross c)
24	9, 20, 23	3bitri 262	. . . . . . . . . . . . . 14 ⊢ (⟨c, ⟨g, b⟩⟩ ∈ ran ( Cross ⊗ (2nd ⊗ 1st )) ↔ ⟨b, g⟩ Cross c)
25	13, 14	brcross 5850	. . . . . . . . . . . . . 14 ⊢ (⟨b, g⟩ Cross c ↔ c = (b × g))
26	8, 24, 25	3bitri 262	. . . . . . . . . . . . 13 ⊢ (⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ↔ c = (b × g))
27	14	otelins2 5792	. . . . . . . . . . . . . 14 ⊢ (⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ Ins2 Ins2 ◡ ≈ ↔ ⟨c, ⟨b, a⟩⟩ ∈ Ins2 ◡ ≈ )
28	13	otelins2 5792	. . . . . . . . . . . . . 14 ⊢ (⟨c, ⟨b, a⟩⟩ ∈ Ins2 ◡ ≈ ↔ ⟨c, a⟩ ∈ ◡ ≈ )
29		df-br 4641	. . . . . . . . . . . . . . 15 ⊢ (c◡ ≈ a ↔ ⟨c, a⟩ ∈ ◡ ≈ )
30		brcnv 4893	. . . . . . . . . . . . . . 15 ⊢ (c◡ ≈ a ↔ a ≈ c)
31	29, 30	bitr3i 242	. . . . . . . . . . . . . 14 ⊢ (⟨c, a⟩ ∈ ◡ ≈ ↔ a ≈ c)
32	27, 28, 31	3bitri 262	. . . . . . . . . . . . 13 ⊢ (⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ Ins2 Ins2 ◡ ≈ ↔ a ≈ c)
33	26, 32	anbi12i 678	. . . . . . . . . . . 12 ⊢ ((⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∧ ⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ Ins2 Ins2 ◡ ≈ ) ↔ (c = (b × g) ∧ a ≈ c))
34	6, 33	bitri 240	. . . . . . . . . . 11 ⊢ (⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) ↔ (c = (b × g) ∧ a ≈ c))
35	34	exbii 1582	. . . . . . . . . 10 ⊢ (∃c⟨c, ⟨g, ⟨b, a⟩⟩⟩ ∈ ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) ↔ ∃c(c = (b × g) ∧ a ≈ c))
36	13, 14	xpex 5116	. . . . . . . . . . 11 ⊢ (b × g) ∈ V
37		breq2 4644	. . . . . . . . . . 11 ⊢ (c = (b × g) → (a ≈ c ↔ a ≈ (b × g)))
38	36, 37	ceqsexv 2895	. . . . . . . . . 10 ⊢ (∃c(c = (b × g) ∧ a ≈ c) ↔ a ≈ (b × g))
39	5, 35, 38	3bitri 262	. . . . . . . . 9 ⊢ (⟨g, ⟨b, a⟩⟩ ∈ ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) ↔ a ≈ (b × g))
40	4, 39	bitri 240	. . . . . . . 8 ⊢ (gran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ )⟨b, a⟩ ↔ a ≈ (b × g))
41	40	rexbii 2640	. . . . . . 7 ⊢ (∃g ∈ N gran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ )⟨b, a⟩ ↔ ∃g ∈ N a ≈ (b × g))
42	2, 3, 41	3bitri 262	. . . . . 6 ⊢ (b(ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N)a ↔ ∃g ∈ N a ≈ (b × g))
43	42	rexbii 2640	. . . . 5 ⊢ (∃b ∈ M b(ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N)a ↔ ∃b ∈ M ∃g ∈ N a ≈ (b × g))
44	1, 43	bitri 240	. . . 4 ⊢ (a ∈ ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N) “ M) ↔ ∃b ∈ M ∃g ∈ N a ≈ (b × g))
45	44	abbi2i 2465	. . 3 ⊢ ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N) “ M) = {a ∣ ∃b ∈ M ∃g ∈ N a ≈ (b × g)}
46		crossex 5851	. . . . . . . . . . 11 ⊢ Cross ∈ V
47		2ndex 5113	. . . . . . . . . . . 12 ⊢ 2nd ∈ V
48		1stex 4740	. . . . . . . . . . . 12 ⊢ 1st ∈ V
49	47, 48	txpex 5786	. . . . . . . . . . 11 ⊢ (2nd ⊗ 1st ) ∈ V
50	46, 49	txpex 5786	. . . . . . . . . 10 ⊢ ( Cross ⊗ (2nd ⊗ 1st )) ∈ V
51	50	rnex 5108	. . . . . . . . 9 ⊢ ran ( Cross ⊗ (2nd ⊗ 1st )) ∈ V
52	51	ins4ex 5800	. . . . . . . 8 ⊢ Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∈ V
53		enex 6032	. . . . . . . . . . 11 ⊢ ≈ ∈ V
54	53	cnvex 5103	. . . . . . . . . 10 ⊢ ◡ ≈ ∈ V
55	54	ins2ex 5798	. . . . . . . . 9 ⊢ Ins2 ◡ ≈ ∈ V
56	55	ins2ex 5798	. . . . . . . 8 ⊢ Ins2 Ins2 ◡ ≈ ∈ V
57	52, 56	inex 4106	. . . . . . 7 ⊢ ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) ∈ V
58	57	rnex 5108	. . . . . 6 ⊢ ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) ∈ V
59		imaexg 4747	. . . . . 6 ⊢ ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) ∈ V ∧ N ∈ NC ) → (ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N) ∈ V)
60	58, 59	mpan 651	. . . . 5 ⊢ (N ∈ NC → (ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N) ∈ V)
61		imaexg 4747	. . . . 5 ⊢ (((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N) ∈ V ∧ M ∈ NC ) → ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N) “ M) ∈ V)
62	60, 61	sylan 457	. . . 4 ⊢ ((N ∈ NC ∧ M ∈ NC ) → ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N) “ M) ∈ V)
63	62	ancoms 439	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ◡ ≈ ) “ N) “ M) ∈ V)
64	45, 63	syl5eqelr 2438	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → {a ∣ ∃b ∈ M ∃g ∈ N a ≈ (b × g)} ∈ V)
65		rexeq 2809	. . . 4 ⊢ (m = M → (∃b ∈ m ∃g ∈ n a ≈ (b × g) ↔ ∃b ∈ M ∃g ∈ n a ≈ (b × g)))
66	65	abbidv 2468	. . 3 ⊢ (m = M → {a ∣ ∃b ∈ m ∃g ∈ n a ≈ (b × g)} = {a ∣ ∃b ∈ M ∃g ∈ n a ≈ (b × g)})
67		rexeq 2809	. . . . 5 ⊢ (n = N → (∃g ∈ n a ≈ (b × g) ↔ ∃g ∈ N a ≈ (b × g)))
68	67	rexbidv 2636	. . . 4 ⊢ (n = N → (∃b ∈ M ∃g ∈ n a ≈ (b × g) ↔ ∃b ∈ M ∃g ∈ N a ≈ (b × g)))
69	68	abbidv 2468	. . 3 ⊢ (n = N → {a ∣ ∃b ∈ M ∃g ∈ n a ≈ (b × g)} = {a ∣ ∃b ∈ M ∃g ∈ N a ≈ (b × g)})
70		df-muc 6103	. . 3 ⊢ ·c = (m ∈ NC , n ∈ NC ↦ {a ∣ ∃b ∈ m ∃g ∈ n a ≈ (b × g)})
71	66, 69, 70	ovmpt2g 5716	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ∧ {a ∣ ∃b ∈ M ∃g ∈ N a ≈ (b × g)} ∈ V) → (M ·c N) = {a ∣ ∃b ∈ M ∃g ∈ N a ≈ (b × g)})
72	64, 71	mpd3an3 1278	1 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M ·c N) = {a ∣ ∃b ∈ M ∃g ∈ N a ≈ (b × g)})

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ∩ cin 3209  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   “ cima 4723   × cxp 4771  ^◡ccnv 4772  ran crn 4774  2^nd c2nd 4784  (class class class)co 5526   ⊗ ctxp 5736   Ins2 cins2 5750   Ins4 cins4 5756   Cross ccross 5764   ≈ cen 6029   NC cncs 6089   ·_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-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-cross 5765  df-en 6030  df-muc 6103

This theorem is referenced by:  mucnc  6132