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 e. Nn /\ N e. Nn /\ (M +o N) =/= (/)) -> Tfin (M +o N) = ( Tfin M +o 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 +o N) =/= (/) <-> E.a a e. (M +o N))
2		nncaddccl 4420	. . . . . . . 8 |- ((M e. Nn /\ N e. Nn ) -> (M +o N) e. Nn )
3		tfincl 4493	. . . . . . . 8 |- ((M +o N) e. Nn -> Tfin (M +o N) e. Nn )
4	2, 3	syl 15	. . . . . . 7 |- ((M e. Nn /\ N e. Nn ) -> Tfin (M +o N) e. Nn )
5	4	3adant3 975	. . . . . 6 |- ((M e. Nn /\ N e. Nn /\ a e. (M +o N)) -> Tfin (M +o N) e. Nn )
6		tfincl 4493	. . . . . . . 8 |- (M e. Nn -> Tfin M e. Nn )
7		tfincl 4493	. . . . . . . 8 |- (N e. Nn -> Tfin N e. Nn )
8		nncaddccl 4420	. . . . . . . 8 |- (( Tfin M e. Nn /\ Tfin N e. Nn ) -> ( Tfin M +o Tfin N) e. Nn )
9	6, 7, 8	syl2an 463	. . . . . . 7 |- ((M e. Nn /\ N e. Nn ) -> ( Tfin M +o Tfin N) e. Nn )
10	9	3adant3 975	. . . . . 6 |- ((M e. Nn /\ N e. Nn /\ a e. (M +o N)) -> ( Tfin M +o Tfin N) e. Nn )
11	2	3adant3 975	. . . . . . 7 |- ((M e. Nn /\ N e. Nn /\ a e. (M +o N)) -> (M +o N) e. Nn )
12		simp3 957	. . . . . . 7 |- ((M e. Nn /\ N e. Nn /\ a e. (M +o N)) -> a e. (M +o N))
13		tfinpw1 4495	. . . . . . 7 |- (((M +o N) e. Nn /\ a e. (M +o N)) -> ~P1 a e. Tfin (M +o N))
14	11, 12, 13	syl2anc 642	. . . . . 6 |- ((M e. Nn /\ N e. Nn /\ a e. (M +o N)) -> ~P1 a e. Tfin (M +o N))
15		eladdc 4399	. . . . . . . 8 |- (a e. (M +o N) <-> E.b e. M E.c e. N ((b i^i c) = (/) /\ a = (b u. c)))
16		simplll 734	. . . . . . . . . . . . 13 |- ((((M e. Nn /\ N e. Nn ) /\ (b e. M /\ c e. N)) /\ (b i^i c) = (/)) -> M e. Nn )
17		simplrl 736	. . . . . . . . . . . . 13 |- ((((M e. Nn /\ N e. Nn ) /\ (b e. M /\ c e. N)) /\ (b i^i c) = (/)) -> b e. M)
18		tfinpw1 4495	. . . . . . . . . . . . 13 |- ((M e. Nn /\ b e. M) -> ~P1 b e. Tfin M)
19	16, 17, 18	syl2anc 642	. . . . . . . . . . . 12 |- ((((M e. Nn /\ N e. Nn ) /\ (b e. M /\ c e. N)) /\ (b i^i c) = (/)) -> ~P1 b e. Tfin M)
20		simpllr 735	. . . . . . . . . . . . 13 |- ((((M e. Nn /\ N e. Nn ) /\ (b e. M /\ c e. N)) /\ (b i^i c) = (/)) -> N e. Nn )
21		simplrr 737	. . . . . . . . . . . . 13 |- ((((M e. Nn /\ N e. Nn ) /\ (b e. M /\ c e. N)) /\ (b i^i c) = (/)) -> c e. N)
22		tfinpw1 4495	. . . . . . . . . . . . 13 |- ((N e. Nn /\ c e. N) -> ~P1 c e. Tfin N)
23	20, 21, 22	syl2anc 642	. . . . . . . . . . . 12 |- ((((M e. Nn /\ N e. Nn ) /\ (b e. M /\ c e. N)) /\ (b i^i c) = (/)) -> ~P1 c e. Tfin N)
24		pw1eq 4144	. . . . . . . . . . . . . 14 |- ((b i^i c) = (/) -> ~P1 (b i^i c) = ~P1 (/))
25		pw1in 4165	. . . . . . . . . . . . . 14 |- ~P1 (b i^i c) = (~P1 b i^i ~P1 c)
26		pw10 4162	. . . . . . . . . . . . . 14 |- ~P1 (/) = (/)
27	24, 25, 26	3eqtr3g 2408	. . . . . . . . . . . . 13 |- ((b i^i c) = (/) -> (~P1 b i^i ~P1 c) = (/))
28	27	adantl 452	. . . . . . . . . . . 12 |- ((((M e. Nn /\ N e. Nn ) /\ (b e. M /\ c e. N)) /\ (b i^i c) = (/)) -> (~P1 b i^i ~P1 c) = (/))
29		eladdci 4400	. . . . . . . . . . . 12 |- ((~P1 b e. Tfin M /\ ~P1 c e. Tfin N /\ (~P1 b i^i ~P1 c) = (/)) -> (~P1 b u. ~P1 c) e. ( Tfin M +o Tfin N))
30	19, 23, 28, 29	syl3anc 1182	. . . . . . . . . . 11 |- ((((M e. Nn /\ N e. Nn ) /\ (b e. M /\ c e. N)) /\ (b i^i c) = (/)) -> (~P1 b u. ~P1 c) e. ( Tfin M +o Tfin N))
31		pw1eq 4144	. . . . . . . . . . . . 13 |- (a = (b u. c) -> ~P1 a = ~P1 (b u. c))
32		pw1un 4164	. . . . . . . . . . . . 13 |- ~P1 (b u. c) = (~P1 b u. ~P1 c)
33	31, 32	syl6eq 2401	. . . . . . . . . . . 12 |- (a = (b u. c) -> ~P1 a = (~P1 b u. ~P1 c))
34	33	eleq1d 2419	. . . . . . . . . . 11 |- (a = (b u. c) -> (~P1 a e. ( Tfin M +o Tfin N) <-> (~P1 b u. ~P1 c) e. ( Tfin M +o Tfin N)))
35	30, 34	syl5ibrcom 213	. . . . . . . . . 10 |- ((((M e. Nn /\ N e. Nn ) /\ (b e. M /\ c e. N)) /\ (b i^i c) = (/)) -> (a = (b u. c) -> ~P1 a e. ( Tfin M +o Tfin N)))
36	35	expimpd 586	. . . . . . . . 9 |- (((M e. Nn /\ N e. Nn ) /\ (b e. M /\ c e. N)) -> (((b i^i c) = (/) /\ a = (b u. c)) -> ~P1 a e. ( Tfin M +o Tfin N)))
37	36	rexlimdvva 2746	. . . . . . . 8 |- ((M e. Nn /\ N e. Nn ) -> (E.b e. M E.c e. N ((b i^i c) = (/) /\ a = (b u. c)) -> ~P1 a e. ( Tfin M +o Tfin N)))
38	15, 37	syl5bi 208	. . . . . . 7 |- ((M e. Nn /\ N e. Nn ) -> (a e. (M +o N) -> ~P1 a e. ( Tfin M +o Tfin N)))
39	38	3impia 1148	. . . . . 6 |- ((M e. Nn /\ N e. Nn /\ a e. (M +o N)) -> ~P1 a e. ( Tfin M +o Tfin N))
40		nnceleq 4431	. . . . . 6 |- ((( Tfin (M +o N) e. Nn /\ ( Tfin M +o Tfin N) e. Nn ) /\ (~P1 a e. Tfin (M +o N) /\ ~P1 a e. ( Tfin M +o Tfin N))) -> Tfin (M +o N) = ( Tfin M +o Tfin N))
41	5, 10, 14, 39, 40	syl22anc 1183	. . . . 5 |- ((M e. Nn /\ N e. Nn /\ a e. (M +o N)) -> Tfin (M +o N) = ( Tfin M +o Tfin N))
42	41	3expia 1153	. . . 4 |- ((M e. Nn /\ N e. Nn ) -> (a e. (M +o N) -> Tfin (M +o N) = ( Tfin M +o Tfin N)))
43	42	exlimdv 1636	. . 3 |- ((M e. Nn /\ N e. Nn ) -> (E.a a e. (M +o N) -> Tfin (M +o N) = ( Tfin M +o Tfin N)))
44	1, 43	syl5bi 208	. 2 |- ((M e. Nn /\ N e. Nn ) -> ((M +o N) =/= (/) -> Tfin (M +o N) = ( Tfin M +o Tfin N)))
45	44	3impia 1148	1 |- ((M e. Nn /\ N e. Nn /\ (M +o N) =/= (/)) -> Tfin (M +o N) = ( Tfin M +o Tfin N))

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616   u. cun 3208   i^i cin 3209  (/)c0 3551  ~P₁ cpw1 4136   Nn cnnc 4374   +o 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