Theorem tfinltfinlem1 4501

Description: Lemma for tfinltfin 4502. Prove the forward direction of the theorem. (Contributed by SF, 2-Feb-2015.)

Assertion

Ref	Expression
tfinltfinlem1	|- ((M e. Nn /\ N e. Nn ) -> (<<M, N>> e. <fin -> << Tfin M, Tfin N>> e. <fin ))

Proof of Theorem tfinltfinlem1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfinnnul 4491	. . . . . 6 |- ((M e. Nn /\ M =/= (/)) -> Tfin M =/= (/))
2	1	ex 423	. . . . 5 |- (M e. Nn -> (M =/= (/) -> Tfin M =/= (/)))
3	2	adantrd 454	. . . 4 |- (M e. Nn -> ((M =/= (/) /\ E.x e. Nn N = ((M +o x) +o 1c)) -> Tfin M =/= (/)))
4	3	adantr 451	. . 3 |- ((M e. Nn /\ N e. Nn ) -> ((M =/= (/) /\ E.x e. Nn N = ((M +o x) +o 1c)) -> Tfin M =/= (/)))
5		addcnul1 4453	. . . . . . . . . . . . . 14 |- (1c +o (/)) = (/)
6		addccom 4407	. . . . . . . . . . . . . 14 |- (1c +o (/)) = ((/) +o 1c)
7	5, 6	eqtr3i 2375	. . . . . . . . . . . . 13 |- (/) = ((/) +o 1c)
8		addceq2 4385	. . . . . . . . . . . . . . . . 17 |- (y = (/) -> ( Tfin M +o y) = ( Tfin M +o (/)))
9		addcnul1 4453	. . . . . . . . . . . . . . . . 17 |- ( Tfin M +o (/)) = (/)
10	8, 9	syl6eq 2401	. . . . . . . . . . . . . . . 16 |- (y = (/) -> ( Tfin M +o y) = (/))
11	10	addceq1d 4390	. . . . . . . . . . . . . . 15 |- (y = (/) -> (( Tfin M +o y) +o 1c) = ((/) +o 1c))
12	11	eqeq2d 2364	. . . . . . . . . . . . . 14 |- (y = (/) -> ((/) = (( Tfin M +o y) +o 1c) <-> (/) = ((/) +o 1c)))
13	12	rspcev 2956	. . . . . . . . . . . . 13 |- (((/) e. Nn /\ (/) = ((/) +o 1c)) -> E.y e. Nn (/) = (( Tfin M +o y) +o 1c))
14	7, 13	mpan2 652	. . . . . . . . . . . 12 |- ((/) e. Nn -> E.y e. Nn (/) = (( Tfin M +o y) +o 1c))
15		eleq1 2413	. . . . . . . . . . . . 13 |- (N = (/) -> (N e. Nn <-> (/) e. Nn ))
16		tfineq 4489	. . . . . . . . . . . . . . . 16 |- (N = (/) -> Tfin N = Tfin (/))
17		tfinnul 4492	. . . . . . . . . . . . . . . 16 |- Tfin (/) = (/)
18	16, 17	syl6eq 2401	. . . . . . . . . . . . . . 15 |- (N = (/) -> Tfin N = (/))
19	18	eqeq1d 2361	. . . . . . . . . . . . . 14 |- (N = (/) -> ( Tfin N = (( Tfin M +o y) +o 1c) <-> (/) = (( Tfin M +o y) +o 1c)))
20	19	rexbidv 2636	. . . . . . . . . . . . 13 |- (N = (/) -> (E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c) <-> E.y e. Nn (/) = (( Tfin M +o y) +o 1c)))
21	15, 20	imbi12d 311	. . . . . . . . . . . 12 |- (N = (/) -> ((N e. Nn -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c)) <-> ((/) e. Nn -> E.y e. Nn (/) = (( Tfin M +o y) +o 1c))))
22	14, 21	mpbiri 224	. . . . . . . . . . 11 |- (N = (/) -> (N e. Nn -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c)))
23	22	adantld 453	. . . . . . . . . 10 |- (N = (/) -> ((M e. Nn /\ N e. Nn ) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c)))
24	23	adantrd 454	. . . . . . . . 9 |- (N = (/) -> (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn )) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c)))
25	24	a1dd 42	. . . . . . . 8 |- (N = (/) -> (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn )) -> (N = ((M +o x) +o 1c) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))))
26		simp2r 982	. . . . . . . . . . . . 13 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> x e. Nn )
27		simp3r 984	. . . . . . . . . . . . . . . . . 18 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> N = ((M +o x) +o 1c))
28		simp3l 983	. . . . . . . . . . . . . . . . . 18 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> N =/= (/))
29	27, 28	eqnetrrd 2537	. . . . . . . . . . . . . . . . 17 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> ((M +o x) +o 1c) =/= (/))
30		addcnnul 4454	. . . . . . . . . . . . . . . . 17 |- (((M +o x) +o 1c) =/= (/) -> ((M +o x) =/= (/) /\ 1c =/= (/)))
31	29, 30	syl 15	. . . . . . . . . . . . . . . 16 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> ((M +o x) =/= (/) /\ 1c =/= (/)))
32	31	simpld 445	. . . . . . . . . . . . . . 15 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> (M +o x) =/= (/))
33		addcnnul 4454	. . . . . . . . . . . . . . 15 |- ((M +o x) =/= (/) -> (M =/= (/) /\ x =/= (/)))
34	32, 33	syl 15	. . . . . . . . . . . . . 14 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> (M =/= (/) /\ x =/= (/)))
35	34	simprd 449	. . . . . . . . . . . . 13 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> x =/= (/))
36		tfinprop 4490	. . . . . . . . . . . . . 14 |- ((x e. Nn /\ x =/= (/)) -> ( Tfin x e. Nn /\ E.y e. x ~P1 y e. Tfin x))
37	36	simpld 445	. . . . . . . . . . . . 13 |- ((x e. Nn /\ x =/= (/)) -> Tfin x e. Nn )
38	26, 35, 37	syl2anc 642	. . . . . . . . . . . 12 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> Tfin x e. Nn )
39		tfineq 4489	. . . . . . . . . . . . . . . 16 |- (N = ((M +o x) +o 1c) -> Tfin N = Tfin ((M +o x) +o 1c))
40	39	adantl 452	. . . . . . . . . . . . . . 15 |- ((N =/= (/) /\ N = ((M +o x) +o 1c)) -> Tfin N = Tfin ((M +o x) +o 1c))
41	40	3ad2ant3 978	. . . . . . . . . . . . . 14 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> Tfin N = Tfin ((M +o x) +o 1c))
42		simp1l 979	. . . . . . . . . . . . . . . 16 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> M e. Nn )
43		nncaddccl 4420	. . . . . . . . . . . . . . . 16 |- ((M e. Nn /\ x e. Nn ) -> (M +o x) e. Nn )
44	42, 26, 43	syl2anc 642	. . . . . . . . . . . . . . 15 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> (M +o x) e. Nn )
45		tfinsuc 4499	. . . . . . . . . . . . . . 15 |- (((M +o x) e. Nn /\ ((M +o x) +o 1c) =/= (/)) -> Tfin ((M +o x) +o 1c) = ( Tfin (M +o x) +o 1c))
46	44, 29, 45	syl2anc 642	. . . . . . . . . . . . . 14 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> Tfin ((M +o x) +o 1c) = ( Tfin (M +o x) +o 1c))
47	41, 46	eqtrd 2385	. . . . . . . . . . . . 13 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> Tfin N = ( Tfin (M +o x) +o 1c))
48		tfindi 4497	. . . . . . . . . . . . . . 15 |- ((M e. Nn /\ x e. Nn /\ (M +o x) =/= (/)) -> Tfin (M +o x) = ( Tfin M +o Tfin x))
49	42, 26, 32, 48	syl3anc 1182	. . . . . . . . . . . . . 14 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> Tfin (M +o x) = ( Tfin M +o Tfin x))
50	49	addceq1d 4390	. . . . . . . . . . . . 13 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> ( Tfin (M +o x) +o 1c) = (( Tfin M +o Tfin x) +o 1c))
51	47, 50	eqtrd 2385	. . . . . . . . . . . 12 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> Tfin N = (( Tfin M +o Tfin x) +o 1c))
52		addceq2 4385	. . . . . . . . . . . . . . 15 |- (y = Tfin x -> ( Tfin M +o y) = ( Tfin M +o Tfin x))
53	52	addceq1d 4390	. . . . . . . . . . . . . 14 |- (y = Tfin x -> (( Tfin M +o y) +o 1c) = (( Tfin M +o Tfin x) +o 1c))
54	53	eqeq2d 2364	. . . . . . . . . . . . 13 |- (y = Tfin x -> ( Tfin N = (( Tfin M +o y) +o 1c) <-> Tfin N = (( Tfin M +o Tfin x) +o 1c)))
55	54	rspcev 2956	. . . . . . . . . . . 12 |- (( Tfin x e. Nn /\ Tfin N = (( Tfin M +o Tfin x) +o 1c)) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))
56	38, 51, 55	syl2anc 642	. . . . . . . . . . 11 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn ) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))
57	56	3expa 1151	. . . . . . . . . 10 |- ((((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn )) /\ (N =/= (/) /\ N = ((M +o x) +o 1c))) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))
58	57	exp32 588	. . . . . . . . 9 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn )) -> (N =/= (/) -> (N = ((M +o x) +o 1c) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))))
59	58	com12 27	. . . . . . . 8 |- (N =/= (/) -> (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn )) -> (N = ((M +o x) +o 1c) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))))
60	25, 59	pm2.61ine 2593	. . . . . . 7 |- (((M e. Nn /\ N e. Nn ) /\ (M =/= (/) /\ x e. Nn )) -> (N = ((M +o x) +o 1c) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c)))
61	60	expr 598	. . . . . 6 |- (((M e. Nn /\ N e. Nn ) /\ M =/= (/)) -> (x e. Nn -> (N = ((M +o x) +o 1c) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))))
62	61	rexlimdv 2738	. . . . 5 |- (((M e. Nn /\ N e. Nn ) /\ M =/= (/)) -> (E.x e. Nn N = ((M +o x) +o 1c) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c)))
63	62	ex 423	. . . 4 |- ((M e. Nn /\ N e. Nn ) -> (M =/= (/) -> (E.x e. Nn N = ((M +o x) +o 1c) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))))
64	63	imp3a 420	. . 3 |- ((M e. Nn /\ N e. Nn ) -> ((M =/= (/) /\ E.x e. Nn N = ((M +o x) +o 1c)) -> E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c)))
65	4, 64	jcad 519	. 2 |- ((M e. Nn /\ N e. Nn ) -> ((M =/= (/) /\ E.x e. Nn N = ((M +o x) +o 1c)) -> ( Tfin M =/= (/) /\ E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))))
66		opkltfing 4450	. 2 |- ((M e. Nn /\ N e. Nn ) -> (<<M, N>> e. <fin <-> (M =/= (/) /\ E.x e. Nn N = ((M +o x) +o 1c))))
67		tfinex 4486	. . . 4 |- Tfin M e. _V
68		tfinex 4486	. . . 4 |- Tfin N e. _V
69		opkltfing 4450	. . . 4 |- (( Tfin M e. _V /\ Tfin N e. _V) -> (<< Tfin M, Tfin N>> e. <fin <-> ( Tfin M =/= (/) /\ E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))))
70	67, 68, 69	mp2an 653	. . 3 |- (<< Tfin M, Tfin N>> e. <fin <-> ( Tfin M =/= (/) /\ E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c)))
71	70	a1i 10	. 2 |- ((M e. Nn /\ N e. Nn ) -> (<< Tfin M, Tfin N>> e. <fin <-> ( Tfin M =/= (/) /\ E.y e. Nn Tfin N = (( Tfin M +o y) +o 1c))))
72	65, 66, 71	3imtr4d 259	1 |- ((M e. Nn /\ N e. Nn ) -> (<<M, N>> e. <fin -> << Tfin M, Tfin N>> e. <fin ))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616  _Vcvv 2860  (/)c0 3551  <<copk 4058  1_cc1c 4135  ~P₁ cpw1 4136   Nn cnnc 4374   +o cplc 4376   <_fin cltfin 4434   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-ltfin 4442  df-tfin 4444

This theorem is referenced by:  tfinltfin  4502