Theorem tfindi 4497

Description: The finite T operation distributes over nonempty cardinal sum. Theorem X.1.32 of [Rosser] p. 529. (Contributed by SF, 26-Jan-2015.)

Assertion

Ref	Expression
tfindi	⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ (M +c N) ≠ ∅) → Tfin (M +c N) = ( Tfin M +c Tfin N))

Proof of Theorem tfindi

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3560	. . 3 ⊢ ((M +c N) ≠ ∅ ↔ ∃a a ∈ (M +c N))
2		nncaddccl 4420	. . . . . . . 8 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (M +c N) ∈ Nn )
3		tfincl 4493	. . . . . . . 8 ⊢ ((M +c N) ∈ Nn → Tfin (M +c N) ∈ Nn )
4	2, 3	syl 15	. . . . . . 7 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → Tfin (M +c N) ∈ Nn )
5	4	3adant3 975	. . . . . 6 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ a ∈ (M +c N)) → Tfin (M +c N) ∈ Nn )
6		tfincl 4493	. . . . . . . 8 ⊢ (M ∈ Nn → Tfin M ∈ Nn )
7		tfincl 4493	. . . . . . . 8 ⊢ (N ∈ Nn → Tfin N ∈ Nn )
8		nncaddccl 4420	. . . . . . . 8 ⊢ (( Tfin M ∈ Nn ∧ Tfin N ∈ Nn ) → ( Tfin M +c Tfin N) ∈ Nn )
9	6, 7, 8	syl2an 463	. . . . . . 7 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → ( Tfin M +c Tfin N) ∈ Nn )
10	9	3adant3 975	. . . . . 6 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ a ∈ (M +c N)) → ( Tfin M +c Tfin N) ∈ Nn )
11	2	3adant3 975	. . . . . . 7 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ a ∈ (M +c N)) → (M +c N) ∈ Nn )
12		simp3 957	. . . . . . 7 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ a ∈ (M +c N)) → a ∈ (M +c N))
13		tfinpw1 4495	. . . . . . 7 ⊢ (((M +c N) ∈ Nn ∧ a ∈ (M +c N)) → ℘1a ∈ Tfin (M +c N))
14	11, 12, 13	syl2anc 642	. . . . . 6 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ a ∈ (M +c N)) → ℘1a ∈ Tfin (M +c N))
15		eladdc 4399	. . . . . . . 8 ⊢ (a ∈ (M +c N) ↔ ∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ a = (b ∪ c)))
16		simplll 734	. . . . . . . . . . . . 13 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ c ∈ N)) ∧ (b ∩ c) = ∅) → M ∈ Nn )
17		simplrl 736	. . . . . . . . . . . . 13 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ c ∈ N)) ∧ (b ∩ c) = ∅) → b ∈ M)
18		tfinpw1 4495	. . . . . . . . . . . . 13 ⊢ ((M ∈ Nn ∧ b ∈ M) → ℘1b ∈ Tfin M)
19	16, 17, 18	syl2anc 642	. . . . . . . . . . . 12 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ c ∈ N)) ∧ (b ∩ c) = ∅) → ℘1b ∈ Tfin M)
20		simpllr 735	. . . . . . . . . . . . 13 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ c ∈ N)) ∧ (b ∩ c) = ∅) → N ∈ Nn )
21		simplrr 737	. . . . . . . . . . . . 13 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ c ∈ N)) ∧ (b ∩ c) = ∅) → c ∈ N)
22		tfinpw1 4495	. . . . . . . . . . . . 13 ⊢ ((N ∈ Nn ∧ c ∈ N) → ℘1c ∈ Tfin N)
23	20, 21, 22	syl2anc 642	. . . . . . . . . . . 12 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ c ∈ N)) ∧ (b ∩ c) = ∅) → ℘1c ∈ Tfin N)
24		pw1eq 4144	. . . . . . . . . . . . . 14 ⊢ ((b ∩ c) = ∅ → ℘1(b ∩ c) = ℘1∅)
25		pw1in 4165	. . . . . . . . . . . . . 14 ⊢ ℘1(b ∩ c) = (℘1b ∩ ℘1c)
26		pw10 4162	. . . . . . . . . . . . . 14 ⊢ ℘1∅ = ∅
27	24, 25, 26	3eqtr3g 2408	. . . . . . . . . . . . 13 ⊢ ((b ∩ c) = ∅ → (℘1b ∩ ℘1c) = ∅)
28	27	adantl 452	. . . . . . . . . . . 12 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ c ∈ N)) ∧ (b ∩ c) = ∅) → (℘1b ∩ ℘1c) = ∅)
29		eladdci 4400	. . . . . . . . . . . 12 ⊢ ((℘1b ∈ Tfin M ∧ ℘1c ∈ Tfin N ∧ (℘1b ∩ ℘1c) = ∅) → (℘1b ∪ ℘1c) ∈ ( Tfin M +c Tfin N))
30	19, 23, 28, 29	syl3anc 1182	. . . . . . . . . . 11 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ c ∈ N)) ∧ (b ∩ c) = ∅) → (℘1b ∪ ℘1c) ∈ ( Tfin M +c Tfin N))
31		pw1eq 4144	. . . . . . . . . . . . 13 ⊢ (a = (b ∪ c) → ℘1a = ℘1(b ∪ c))
32		pw1un 4164	. . . . . . . . . . . . 13 ⊢ ℘1(b ∪ c) = (℘1b ∪ ℘1c)
33	31, 32	syl6eq 2401	. . . . . . . . . . . 12 ⊢ (a = (b ∪ c) → ℘1a = (℘1b ∪ ℘1c))
34	33	eleq1d 2419	. . . . . . . . . . 11 ⊢ (a = (b ∪ c) → (℘1a ∈ ( Tfin M +c Tfin N) ↔ (℘1b ∪ ℘1c) ∈ ( Tfin M +c Tfin N)))
35	30, 34	syl5ibrcom 213	. . . . . . . . . 10 ⊢ ((((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ c ∈ N)) ∧ (b ∩ c) = ∅) → (a = (b ∪ c) → ℘1a ∈ ( Tfin M +c Tfin N)))
36	35	expimpd 586	. . . . . . . . 9 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ (b ∈ M ∧ c ∈ N)) → (((b ∩ c) = ∅ ∧ a = (b ∪ c)) → ℘1a ∈ ( Tfin M +c Tfin N)))
37	36	rexlimdvva 2746	. . . . . . . 8 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (∃b ∈ M ∃c ∈ N ((b ∩ c) = ∅ ∧ a = (b ∪ c)) → ℘1a ∈ ( Tfin M +c Tfin N)))
38	15, 37	syl5bi 208	. . . . . . 7 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (a ∈ (M +c N) → ℘1a ∈ ( Tfin M +c Tfin N)))
39	38	3impia 1148	. . . . . 6 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ a ∈ (M +c N)) → ℘1a ∈ ( Tfin M +c Tfin N))
40		nnceleq 4431	. . . . . 6 ⊢ ((( Tfin (M +c N) ∈ Nn ∧ ( Tfin M +c Tfin N) ∈ Nn ) ∧ (℘1a ∈ Tfin (M +c N) ∧ ℘1a ∈ ( Tfin M +c Tfin N))) → Tfin (M +c N) = ( Tfin M +c Tfin N))
41	5, 10, 14, 39, 40	syl22anc 1183	. . . . 5 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ a ∈ (M +c N)) → Tfin (M +c N) = ( Tfin M +c Tfin N))
42	41	3expia 1153	. . . 4 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (a ∈ (M +c N) → Tfin (M +c N) = ( Tfin M +c Tfin N)))
43	42	exlimdv 1636	. . 3 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → (∃a a ∈ (M +c N) → Tfin (M +c N) = ( Tfin M +c Tfin N)))
44	1, 43	syl5bi 208	. 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ) → ((M +c N) ≠ ∅ → Tfin (M +c N) = ( Tfin M +c Tfin N)))
45	44	3impia 1148	1 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ (M +c N) ≠ ∅) → Tfin (M +c N) = ( Tfin M +c Tfin N))

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  ℘₁cpw1 4136   Nn cnnc 4374   +_c cplc 4376   T_fin ctfin 4436

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-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-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-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  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-tfin 4444

This theorem is referenced by:  tfinltfinlem1  4501  eventfin  4518  oddtfin  4519  sfintfin  4533