Theorem nchoicelem12 6301

Description: Lemma for nchoice 6309. If the T-raising of a cardinal yields a finite special set, then so does the initial set. Theorem 7.1 of [Specker] p. 974. (Contributed by SF, 18-Mar-2015.)

Assertion

Ref	Expression
nchoicelem12	|- ((M e. NC /\ ( Spac ` Tc M) e. Fin ) -> ( Spac ` M) e. Fin )

Proof of Theorem nchoicelem12

Dummy variables m t x k n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		finnc 6244	. . . 4 |- (( Spac ` Tc M) e. Fin <-> Nc ( Spac ` Tc M) e. Nn )
2		risset 2662	. . . 4 |- ( Nc ( Spac ` Tc M) e. Nn <-> E.x e. Nn x = Nc ( Spac ` Tc M))
3	1, 2	bitri 240	. . 3 |- (( Spac ` Tc M) e. Fin <-> E.x e. Nn x = Nc ( Spac ` Tc M))
4		nchoicelem11 6300	. . . . . . 7 |- {t | A.m e. NC (t = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )} e. _V
5		eqeq1 2359	. . . . . . . . 9 |- (t = 0c -> (t = Nc ( Spac ` Tc m) <-> 0c = Nc ( Spac ` Tc m)))
6	5	imbi1d 308	. . . . . . . 8 |- (t = 0c -> ((t = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> (0c = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )))
7	6	ralbidv 2635	. . . . . . 7 |- (t = 0c -> (A.m e. NC (t = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> A.m e. NC (0c = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )))
8		eqeq1 2359	. . . . . . . . 9 |- (t = n -> (t = Nc ( Spac ` Tc m) <-> n = Nc ( Spac ` Tc m)))
9	8	imbi1d 308	. . . . . . . 8 |- (t = n -> ((t = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )))
10	9	ralbidv 2635	. . . . . . 7 |- (t = n -> (A.m e. NC (t = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )))
11		eqeq1 2359	. . . . . . . . . 10 |- (t = (n +o 1c) -> (t = Nc ( Spac ` Tc m) <-> (n +o 1c) = Nc ( Spac ` Tc m)))
12	11	imbi1d 308	. . . . . . . . 9 |- (t = (n +o 1c) -> ((t = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> ((n +o 1c) = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )))
13	12	ralbidv 2635	. . . . . . . 8 |- (t = (n +o 1c) -> (A.m e. NC (t = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> A.m e. NC ((n +o 1c) = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )))
14		tceq 6159	. . . . . . . . . . . . 13 |- (m = k -> Tc m = Tc k)
15	14	fveq2d 5333	. . . . . . . . . . . 12 |- (m = k -> ( Spac ` Tc m) = ( Spac ` Tc k))
16	15	nceqd 6111	. . . . . . . . . . 11 |- (m = k -> Nc ( Spac ` Tc m) = Nc ( Spac ` Tc k))
17	16	eqeq2d 2364	. . . . . . . . . 10 |- (m = k -> ((n +o 1c) = Nc ( Spac ` Tc m) <-> (n +o 1c) = Nc ( Spac ` Tc k)))
18		fveq2 5329	. . . . . . . . . . . 12 |- (m = k -> ( Spac ` m) = ( Spac ` k))
19	18	nceqd 6111	. . . . . . . . . . 11 |- (m = k -> Nc ( Spac ` m) = Nc ( Spac ` k))
20	19	eleq1d 2419	. . . . . . . . . 10 |- (m = k -> ( Nc ( Spac ` m) e. Nn <-> Nc ( Spac ` k) e. Nn ))
21	17, 20	imbi12d 311	. . . . . . . . 9 |- (m = k -> (((n +o 1c) = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn )))
22	21	cbvralv 2836	. . . . . . . 8 |- (A.m e. NC ((n +o 1c) = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> A.k e. NC ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn ))
23	13, 22	syl6bb 252	. . . . . . 7 |- (t = (n +o 1c) -> (A.m e. NC (t = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> A.k e. NC ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn )))
24		eqeq1 2359	. . . . . . . . 9 |- (t = x -> (t = Nc ( Spac ` Tc m) <-> x = Nc ( Spac ` Tc m)))
25	24	imbi1d 308	. . . . . . . 8 |- (t = x -> ((t = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> (x = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )))
26	25	ralbidv 2635	. . . . . . 7 |- (t = x -> (A.m e. NC (t = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> A.m e. NC (x = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )))
27		tccl 6161	. . . . . . . . . . . . 13 |- (m e. NC -> Tc m e. NC )
28		te0c 6238	. . . . . . . . . . . . 13 |- (m e. NC -> ( Tc m ↑c 0c) e. NC )
29		nchoicelem7 6296	. . . . . . . . . . . . 13 |- (( Tc m e. NC /\ ( Tc m ↑c 0c) e. NC ) -> Nc ( Spac ` Tc m) = ( Nc ( Spac ` (2c ↑c Tc m)) +o 1c))
30	27, 28, 29	syl2anc 642	. . . . . . . . . . . 12 |- (m e. NC -> Nc ( Spac ` Tc m) = ( Nc ( Spac ` (2c ↑c Tc m)) +o 1c))
31		0cnsuc 4402	. . . . . . . . . . . . 13 |- ( Nc ( Spac ` (2c ↑c Tc m)) +o 1c) =/= 0c
32	31	a1i 10	. . . . . . . . . . . 12 |- (m e. NC -> ( Nc ( Spac ` (2c ↑c Tc m)) +o 1c) =/= 0c)
33	30, 32	eqnetrd 2535	. . . . . . . . . . 11 |- (m e. NC -> Nc ( Spac ` Tc m) =/= 0c)
34	33	necomd 2600	. . . . . . . . . 10 |- (m e. NC -> 0c =/= Nc ( Spac ` Tc m))
35		df-ne 2519	. . . . . . . . . 10 |- (0c =/= Nc ( Spac ` Tc m) <-> -. 0c = Nc ( Spac ` Tc m))
36	34, 35	sylib 188	. . . . . . . . 9 |- (m e. NC -> -. 0c = Nc ( Spac ` Tc m))
37	36	pm2.21d 98	. . . . . . . 8 |- (m e. NC -> (0c = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ))
38	37	rgen 2680	. . . . . . 7 |- A.m e. NC (0c = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )
39		2nnc 6168	. . . . . . . . . . . . . . . . 17 |- 2c e. Nn
40		ceclnn1 6190	. . . . . . . . . . . . . . . . 17 |- ((2c e. Nn /\ k e. NC /\ (k ↑c 0c) e. NC ) -> (2c ↑c k) e. NC )
41	39, 40	mp3an1 1264	. . . . . . . . . . . . . . . 16 |- ((k e. NC /\ (k ↑c 0c) e. NC ) -> (2c ↑c k) e. NC )
42		tceq 6159	. . . . . . . . . . . . . . . . . . . . 21 |- (m = (2c ↑c k) -> Tc m = Tc (2c ↑c k))
43	42	fveq2d 5333	. . . . . . . . . . . . . . . . . . . 20 |- (m = (2c ↑c k) -> ( Spac ` Tc m) = ( Spac ` Tc (2c ↑c k)))
44	43	nceqd 6111	. . . . . . . . . . . . . . . . . . 19 |- (m = (2c ↑c k) -> Nc ( Spac ` Tc m) = Nc ( Spac ` Tc (2c ↑c k)))
45	44	eqeq2d 2364	. . . . . . . . . . . . . . . . . 18 |- (m = (2c ↑c k) -> (n = Nc ( Spac ` Tc m) <-> n = Nc ( Spac ` Tc (2c ↑c k))))
46		fveq2 5329	. . . . . . . . . . . . . . . . . . . 20 |- (m = (2c ↑c k) -> ( Spac ` m) = ( Spac ` (2c ↑c k)))
47	46	nceqd 6111	. . . . . . . . . . . . . . . . . . 19 |- (m = (2c ↑c k) -> Nc ( Spac ` m) = Nc ( Spac ` (2c ↑c k)))
48	47	eleq1d 2419	. . . . . . . . . . . . . . . . . 18 |- (m = (2c ↑c k) -> ( Nc ( Spac ` m) e. Nn <-> Nc ( Spac ` (2c ↑c k)) e. Nn ))
49	45, 48	imbi12d 311	. . . . . . . . . . . . . . . . 17 |- (m = (2c ↑c k) -> ((n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> (n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn )))
50	49	rspcv 2952	. . . . . . . . . . . . . . . 16 |- ((2c ↑c k) e. NC -> (A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) -> (n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn )))
51	41, 50	syl 15	. . . . . . . . . . . . . . 15 |- ((k e. NC /\ (k ↑c 0c) e. NC ) -> (A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) -> (n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn )))
52	51	ancoms 439	. . . . . . . . . . . . . 14 |- (((k ↑c 0c) e. NC /\ k e. NC ) -> (A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) -> (n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn )))
53	52	adantrl 696	. . . . . . . . . . . . 13 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> (A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) -> (n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn )))
54		tccl 6161	. . . . . . . . . . . . . . . . . . . . . . 23 |- (k e. NC -> Tc k e. NC )
55		te0c 6238	. . . . . . . . . . . . . . . . . . . . . . 23 |- (k e. NC -> ( Tc k ↑c 0c) e. NC )
56		nchoicelem7 6296	. . . . . . . . . . . . . . . . . . . . . . 23 |- (( Tc k e. NC /\ ( Tc k ↑c 0c) e. NC ) -> Nc ( Spac ` Tc k) = ( Nc ( Spac ` (2c ↑c Tc k)) +o 1c))
57	54, 55, 56	syl2anc 642	. . . . . . . . . . . . . . . . . . . . . 22 |- (k e. NC -> Nc ( Spac ` Tc k) = ( Nc ( Spac ` (2c ↑c Tc k)) +o 1c))
58	57	adantl 452	. . . . . . . . . . . . . . . . . . . . 21 |- ((n e. Nn /\ k e. NC ) -> Nc ( Spac ` Tc k) = ( Nc ( Spac ` (2c ↑c Tc k)) +o 1c))
59	58	adantl 452	. . . . . . . . . . . . . . . . . . . 20 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> Nc ( Spac ` Tc k) = ( Nc ( Spac ` (2c ↑c Tc k)) +o 1c))
60	59	eqeq2d 2364	. . . . . . . . . . . . . . . . . . 19 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> ((n +o 1c) = Nc ( Spac ` Tc k) <-> (n +o 1c) = ( Nc ( Spac ` (2c ↑c Tc k)) +o 1c)))
61		nnnc 6147	. . . . . . . . . . . . . . . . . . . . . . 23 |- (n e. Nn -> n e. NC )
62	61	adantr 451	. . . . . . . . . . . . . . . . . . . . . 22 |- ((n e. Nn /\ k e. NC ) -> n e. NC )
63	62	adantl 452	. . . . . . . . . . . . . . . . . . . . 21 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> n e. NC )
64		fvex 5340	. . . . . . . . . . . . . . . . . . . . . 22 |- ( Spac ` (2c ↑c Tc k)) e. _V
65	64	ncelncsi 6122	. . . . . . . . . . . . . . . . . . . . 21 |- Nc ( Spac ` (2c ↑c Tc k)) e. NC
66		peano4nc 6151	. . . . . . . . . . . . . . . . . . . . 21 |- ((n e. NC /\ Nc ( Spac ` (2c ↑c Tc k)) e. NC ) -> ((n +o 1c) = ( Nc ( Spac ` (2c ↑c Tc k)) +o 1c) <-> n = Nc ( Spac ` (2c ↑c Tc k))))
67	63, 65, 66	sylancl 643	. . . . . . . . . . . . . . . . . . . 20 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> ((n +o 1c) = ( Nc ( Spac ` (2c ↑c Tc k)) +o 1c) <-> n = Nc ( Spac ` (2c ↑c Tc k))))
68		tce2 6237	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((k e. NC /\ (k ↑c 0c) e. NC ) -> Tc (2c ↑c k) = (2c ↑c Tc k))
69	68	ancoms 439	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((k ↑c 0c) e. NC /\ k e. NC ) -> Tc (2c ↑c k) = (2c ↑c Tc k))
70	69	adantrl 696	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> Tc (2c ↑c k) = (2c ↑c Tc k))
71	70	fveq2d 5333	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> ( Spac ` Tc (2c ↑c k)) = ( Spac ` (2c ↑c Tc k)))
72	71	nceqd 6111	. . . . . . . . . . . . . . . . . . . . . 22 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> Nc ( Spac ` Tc (2c ↑c k)) = Nc ( Spac ` (2c ↑c Tc k)))
73	72	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . 21 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> (n = Nc ( Spac ` Tc (2c ↑c k)) <-> n = Nc ( Spac ` (2c ↑c Tc k))))
74	73	biimprd 214	. . . . . . . . . . . . . . . . . . . 20 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> (n = Nc ( Spac ` (2c ↑c Tc k)) -> n = Nc ( Spac ` Tc (2c ↑c k))))
75	67, 74	sylbid 206	. . . . . . . . . . . . . . . . . . 19 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> ((n +o 1c) = ( Nc ( Spac ` (2c ↑c Tc k)) +o 1c) -> n = Nc ( Spac ` Tc (2c ↑c k))))
76	60, 75	sylbid 206	. . . . . . . . . . . . . . . . . 18 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> n = Nc ( Spac ` Tc (2c ↑c k))))
77	76	imim1d 69	. . . . . . . . . . . . . . . . 17 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> ((n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn ) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` (2c ↑c k)) e. Nn )))
78	77	imp 418	. . . . . . . . . . . . . . . 16 |- ((((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) /\ (n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn )) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` (2c ↑c k)) e. Nn ))
79		peano2 4404	. . . . . . . . . . . . . . . 16 |- ( Nc ( Spac ` (2c ↑c k)) e. Nn -> ( Nc ( Spac ` (2c ↑c k)) +o 1c) e. Nn )
80	78, 79	syl6 29	. . . . . . . . . . . . . . 15 |- ((((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) /\ (n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn )) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> ( Nc ( Spac ` (2c ↑c k)) +o 1c) e. Nn ))
81		nchoicelem7 6296	. . . . . . . . . . . . . . . . . . 19 |- ((k e. NC /\ (k ↑c 0c) e. NC ) -> Nc ( Spac ` k) = ( Nc ( Spac ` (2c ↑c k)) +o 1c))
82	81	ancoms 439	. . . . . . . . . . . . . . . . . 18 |- (((k ↑c 0c) e. NC /\ k e. NC ) -> Nc ( Spac ` k) = ( Nc ( Spac ` (2c ↑c k)) +o 1c))
83	82	adantrl 696	. . . . . . . . . . . . . . . . 17 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> Nc ( Spac ` k) = ( Nc ( Spac ` (2c ↑c k)) +o 1c))
84	83	adantr 451	. . . . . . . . . . . . . . . 16 |- ((((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) /\ (n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn )) -> Nc ( Spac ` k) = ( Nc ( Spac ` (2c ↑c k)) +o 1c))
85	84	eleq1d 2419	. . . . . . . . . . . . . . 15 |- ((((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) /\ (n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn )) -> ( Nc ( Spac ` k) e. Nn <-> ( Nc ( Spac ` (2c ↑c k)) +o 1c) e. Nn ))
86	80, 85	sylibrd 225	. . . . . . . . . . . . . 14 |- ((((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) /\ (n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn )) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn ))
87	86	ex 423	. . . . . . . . . . . . 13 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> ((n = Nc ( Spac ` Tc (2c ↑c k)) -> Nc ( Spac ` (2c ↑c k)) e. Nn ) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn )))
88	53, 87	syld 40	. . . . . . . . . . . 12 |- (((k ↑c 0c) e. NC /\ (n e. Nn /\ k e. NC )) -> (A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn )))
89	88	expimpd 586	. . . . . . . . . . 11 |- ((k ↑c 0c) e. NC -> (((n e. Nn /\ k e. NC ) /\ A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn )))
90		nchoicelem3 6292	. . . . . . . . . . . . . . . . 17 |- ((k e. NC /\ -. (k ↑c 0c) e. NC ) -> ( Spac ` k) = {k})
91	90	nceqd 6111	. . . . . . . . . . . . . . . 16 |- ((k e. NC /\ -. (k ↑c 0c) e. NC ) -> Nc ( Spac ` k) = Nc {k})
92		vex 2863	. . . . . . . . . . . . . . . . . 18 |- k e. _V
93	92	df1c3 6141	. . . . . . . . . . . . . . . . 17 |- 1c = Nc {k}
94		1cnnc 4409	. . . . . . . . . . . . . . . . 17 |- 1c e. Nn
95	93, 94	eqeltrri 2424	. . . . . . . . . . . . . . . 16 |- Nc {k} e. Nn
96	91, 95	syl6eqel 2441	. . . . . . . . . . . . . . 15 |- ((k e. NC /\ -. (k ↑c 0c) e. NC ) -> Nc ( Spac ` k) e. Nn )
97	96	a1d 22	. . . . . . . . . . . . . 14 |- ((k e. NC /\ -. (k ↑c 0c) e. NC ) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn ))
98	97	expcom 424	. . . . . . . . . . . . 13 |- (-. (k ↑c 0c) e. NC -> (k e. NC -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn )))
99	98	adantld 453	. . . . . . . . . . . 12 |- (-. (k ↑c 0c) e. NC -> ((n e. Nn /\ k e. NC ) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn )))
100	99	adantrd 454	. . . . . . . . . . 11 |- (-. (k ↑c 0c) e. NC -> (((n e. Nn /\ k e. NC ) /\ A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn )))
101	89, 100	pm2.61i 156	. . . . . . . . . 10 |- (((n e. Nn /\ k e. NC ) /\ A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn ))
102	101	an32s 779	. . . . . . . . 9 |- (((n e. Nn /\ A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )) /\ k e. NC ) -> ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn ))
103	102	ralrimiva 2698	. . . . . . . 8 |- ((n e. Nn /\ A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn )) -> A.k e. NC ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn ))
104	103	ex 423	. . . . . . 7 |- (n e. Nn -> (A.m e. NC (n = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) -> A.k e. NC ((n +o 1c) = Nc ( Spac ` Tc k) -> Nc ( Spac ` k) e. Nn )))
105	4, 7, 10, 23, 26, 38, 104	finds 4412	. . . . . 6 |- (x e. Nn -> A.m e. NC (x = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ))
106		tceq 6159	. . . . . . . . . . 11 |- (m = M -> Tc m = Tc M)
107	106	fveq2d 5333	. . . . . . . . . 10 |- (m = M -> ( Spac ` Tc m) = ( Spac ` Tc M))
108	107	nceqd 6111	. . . . . . . . 9 |- (m = M -> Nc ( Spac ` Tc m) = Nc ( Spac ` Tc M))
109	108	eqeq2d 2364	. . . . . . . 8 |- (m = M -> (x = Nc ( Spac ` Tc m) <-> x = Nc ( Spac ` Tc M)))
110		fveq2 5329	. . . . . . . . . . 11 |- (m = M -> ( Spac ` m) = ( Spac ` M))
111	110	nceqd 6111	. . . . . . . . . 10 |- (m = M -> Nc ( Spac ` m) = Nc ( Spac ` M))
112	111	eleq1d 2419	. . . . . . . . 9 |- (m = M -> ( Nc ( Spac ` m) e. Nn <-> Nc ( Spac ` M) e. Nn ))
113		finnc 6244	. . . . . . . . 9 |- (( Spac ` M) e. Fin <-> Nc ( Spac ` M) e. Nn )
114	112, 113	syl6bbr 254	. . . . . . . 8 |- (m = M -> ( Nc ( Spac ` m) e. Nn <-> ( Spac ` M) e. Fin ))
115	109, 114	imbi12d 311	. . . . . . 7 |- (m = M -> ((x = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) <-> (x = Nc ( Spac ` Tc M) -> ( Spac ` M) e. Fin )))
116	115	rspccv 2953	. . . . . 6 |- (A.m e. NC (x = Nc ( Spac ` Tc m) -> Nc ( Spac ` m) e. Nn ) -> (M e. NC -> (x = Nc ( Spac ` Tc M) -> ( Spac ` M) e. Fin )))
117	105, 116	syl 15	. . . . 5 |- (x e. Nn -> (M e. NC -> (x = Nc ( Spac ` Tc M) -> ( Spac ` M) e. Fin )))
118	117	com23 72	. . . 4 |- (x e. Nn -> (x = Nc ( Spac ` Tc M) -> (M e. NC -> ( Spac ` M) e. Fin )))
119	118	rexlimiv 2733	. . 3 |- (E.x e. Nn x = Nc ( Spac ` Tc M) -> (M e. NC -> ( Spac ` M) e. Fin ))
120	3, 119	sylbi 187	. 2 |- (( Spac ` Tc M) e. Fin -> (M e. NC -> ( Spac ` M) e. Fin ))
121	120	impcom 419	1 |- ((M e. NC /\ ( Spac ` Tc M) e. Fin ) -> ( Spac ` M) e. Fin )

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  A.wral 2615  E.wrex 2616  {csn 3738  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376   Fin cfin 4377  ` cfv 4782  (class class class)co 5526   NC cncs 6089   Nc cnc 6092   T_c ctc 6094  2_cc2c 6095   ↑_c cce 6097   Sp_ac cspac 6274

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-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-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-fix 5741  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-fullfun 5769  df-clos1 5874  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-lec 6100  df-ltc 6101  df-nc 6102  df-tc 6104  df-2c 6105  df-ce 6107  df-tcfn 6108  df-spac 6275

This theorem is referenced by:  nchoicelem19  6308