Theorem axlowdimlem13 25834

Description: Lemma for axlowdim 25841. Establish that P and Q are different points. (Contributed by Scott Fenton, 21-Apr-2013.)

Hypotheses

Ref	Expression
axlowdimlem13.1	|- P = ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) )
axlowdimlem13.2	|- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )

Assertion

Ref	Expression
axlowdimlem13	|- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> P =/= Q )

Proof of Theorem axlowdimlem13

Step	Hyp	Ref	Expression
1		2ne0 11113	. . . . . . . . 9 |- 2 =/= 0
2	1	neii 2796	. . . . . . . 8 |- -. 2 = 0
3		eqcom 2629	. . . . . . . . 9 |- ( 2 = 0 <-> 0 = 2 )
4		1pneg1e0 11129	. . . . . . . . . . 11 |- ( 1 + -u 1 ) = 0
5	4	eqcomi 2631	. . . . . . . . . 10 |- 0 = ( 1 + -u 1 )
6		df-2 11079	. . . . . . . . . 10 |- 2 = ( 1 + 1 )
7	5, 6	eqeq12i 2636	. . . . . . . . 9 |- ( 0 = 2 <-> ( 1 + -u 1 ) = ( 1 + 1 ) )
8		ax-1cn 9994	. . . . . . . . . 10 |- 1 e. CC
9		neg1cn 11124	. . . . . . . . . 10 |- -u 1 e. CC
10	8, 9, 8	addcani 10229	. . . . . . . . 9 |- ( ( 1 + -u 1 ) = ( 1 + 1 ) <-> -u 1 = 1 )
11	3, 7, 10	3bitri 286	. . . . . . . 8 |- ( 2 = 0 <-> -u 1 = 1 )
12	2, 11	mtbi 312	. . . . . . 7 |- -. -u 1 = 1
13	12	intnanr 961	. . . . . 6 |- -. ( -u 1 = 1 /\ 0 = 0 )
14		ax-1ne0 10005	. . . . . . . . 9 |- 1 =/= 0
15	14	neii 2796	. . . . . . . 8 |- -. 1 = 0
16		negeq0 10335	. . . . . . . . 9 |- ( 1 e. CC -> ( 1 = 0 <-> -u 1 = 0 ) )
17	8, 16	ax-mp 5	. . . . . . . 8 |- ( 1 = 0 <-> -u 1 = 0 )
18	15, 17	mtbi 312	. . . . . . 7 |- -. -u 1 = 0
19	18	intnanr 961	. . . . . 6 |- -. ( -u 1 = 0 /\ 0 = 1 )
20	13, 19	pm3.2ni 899	. . . . 5 |- -. ( ( -u 1 = 1 /\ 0 = 0 ) \/ ( -u 1 = 0 /\ 0 = 1 ) )
21		negex 10279	. . . . . 6 |- -u 1 e. _V
22		c0ex 10034	. . . . . 6 |- 0 e. _V
23		1ex 10035	. . . . . 6 |- 1 e. _V
24	21, 22, 23, 22	preq12b 4382	. . . . 5 |- ( { -u 1 , 0 } = { 1 , 0 } <-> ( ( -u 1 = 1 /\ 0 = 0 ) \/ ( -u 1 = 0 /\ 0 = 1 ) ) )
25	20, 24	mtbir 313	. . . 4 |- -. { -u 1 , 0 } = { 1 , 0 }
26		3ex 11096	. . . . . . . . 9 |- 3 e. _V
27	26	rnsnop 5616	. . . . . . . 8 |- ran { <. 3 , -u 1 >. } = { -u 1 }
28	27	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ran { <. 3 , -u 1 >. } = { -u 1 } )
29		elnnuz 11724	. . . . . . . . . . . 12 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
30		eluzfz1 12348	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... N ) )
31	29, 30	sylbi 207	. . . . . . . . . . 11 |- ( N e. NN -> 1 e. ( 1 ... N ) )
32		df-3 11080	. . . . . . . . . . . . . . . 16 |- 3 = ( 2 + 1 )
33		1e0p1 11552	. . . . . . . . . . . . . . . 16 |- 1 = ( 0 + 1 )
34	32, 33	eqeq12i 2636	. . . . . . . . . . . . . . 15 |- ( 3 = 1 <-> ( 2 + 1 ) = ( 0 + 1 ) )
35		2cn 11091	. . . . . . . . . . . . . . . 16 |- 2 e. CC
36		0cn 10032	. . . . . . . . . . . . . . . 16 |- 0 e. CC
37	35, 36, 8	addcan2i 10230	. . . . . . . . . . . . . . 15 |- ( ( 2 + 1 ) = ( 0 + 1 ) <-> 2 = 0 )
38	34, 37	bitri 264	. . . . . . . . . . . . . 14 |- ( 3 = 1 <-> 2 = 0 )
39	38	necon3bii 2846	. . . . . . . . . . . . 13 |- ( 3 =/= 1 <-> 2 =/= 0 )
40	1, 39	mpbir 221	. . . . . . . . . . . 12 |- 3 =/= 1
41	40	necomi 2848	. . . . . . . . . . 11 |- 1 =/= 3
42	31, 41	jctir 561	. . . . . . . . . 10 |- ( N e. NN -> ( 1 e. ( 1 ... N ) /\ 1 =/= 3 ) )
43		eldifsn 4317	. . . . . . . . . 10 |- ( 1 e. ( ( 1 ... N ) \ { 3 } ) <-> ( 1 e. ( 1 ... N ) /\ 1 =/= 3 ) )
44	42, 43	sylibr 224	. . . . . . . . 9 |- ( N e. NN -> 1 e. ( ( 1 ... N ) \ { 3 } ) )
45	44	adantr 481	. . . . . . . 8 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> 1 e. ( ( 1 ... N ) \ { 3 } ) )
46		ne0i 3921	. . . . . . . 8 |- ( 1 e. ( ( 1 ... N ) \ { 3 } ) -> ( ( 1 ... N ) \ { 3 } ) =/= (/) )
47		rnxp 5564	. . . . . . . 8 |- ( ( ( 1 ... N ) \ { 3 } ) =/= (/) -> ran ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) = { 0 } )
48	45, 46, 47	3syl 18	. . . . . . 7 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ran ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) = { 0 } )
49	28, 48	uneq12d 3768	. . . . . 6 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ( ran { <. 3 , -u 1 >. } u. ran ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) = ( { -u 1 } u. { 0 } ) )
50		rnun 5541	. . . . . 6 |- ran ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) = ( ran { <. 3 , -u 1 >. } u. ran ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) )
51		df-pr 4180	. . . . . 6 |- { -u 1 , 0 } = ( { -u 1 } u. { 0 } )
52	49, 50, 51	3eqtr4g 2681	. . . . 5 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ran ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) = { -u 1 , 0 } )
53		ovex 6678	. . . . . . . . 9 |- ( I + 1 ) e. _V
54	53	rnsnop 5616	. . . . . . . 8 |- ran { <. ( I + 1 ) , 1 >. } = { 1 }
55	54	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ran { <. ( I + 1 ) , 1 >. } = { 1 } )
56		nnz 11399	. . . . . . . . . . 11 |- ( N e. NN -> N e. ZZ )
57		fzssp1 12384	. . . . . . . . . . . 12 |- ( 1 ... ( N - 1 ) ) C_ ( 1 ... ( ( N - 1 ) + 1 ) )
58		zcn 11382	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. CC )
59		npcan1 10455	. . . . . . . . . . . . . 14 |- ( N e. CC -> ( ( N - 1 ) + 1 ) = N )
60	59	oveq2d 6666	. . . . . . . . . . . . 13 |- ( N e. CC -> ( 1 ... ( ( N - 1 ) + 1 ) ) = ( 1 ... N ) )
61	58, 60	syl 17	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( 1 ... ( ( N - 1 ) + 1 ) ) = ( 1 ... N ) )
62	57, 61	syl5sseq 3653	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 1 ... ( N - 1 ) ) C_ ( 1 ... N ) )
63	56, 62	syl 17	. . . . . . . . . 10 |- ( N e. NN -> ( 1 ... ( N - 1 ) ) C_ ( 1 ... N ) )
64	63	sselda 3603	. . . . . . . . 9 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> I e. ( 1 ... N ) )
65		elfzelz 12342	. . . . . . . . . . . 12 |- ( I e. ( 1 ... ( N - 1 ) ) -> I e. ZZ )
66	65	zred 11482	. . . . . . . . . . 11 |- ( I e. ( 1 ... ( N - 1 ) ) -> I e. RR )
67		id 22	. . . . . . . . . . . 12 |- ( I e. RR -> I e. RR )
68		ltp1 10861	. . . . . . . . . . . 12 |- ( I e. RR -> I < ( I + 1 ) )
69	67, 68	ltned 10173	. . . . . . . . . . 11 |- ( I e. RR -> I =/= ( I + 1 ) )
70	66, 69	syl 17	. . . . . . . . . 10 |- ( I e. ( 1 ... ( N - 1 ) ) -> I =/= ( I + 1 ) )
71	70	adantl 482	. . . . . . . . 9 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> I =/= ( I + 1 ) )
72		eldifsn 4317	. . . . . . . . 9 |- ( I e. ( ( 1 ... N ) \ { ( I + 1 ) } ) <-> ( I e. ( 1 ... N ) /\ I =/= ( I + 1 ) ) )
73	64, 71, 72	sylanbrc 698	. . . . . . . 8 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> I e. ( ( 1 ... N ) \ { ( I + 1 ) } ) )
74		ne0i 3921	. . . . . . . 8 |- ( I e. ( ( 1 ... N ) \ { ( I + 1 ) } ) -> ( ( 1 ... N ) \ { ( I + 1 ) } ) =/= (/) )
75		rnxp 5564	. . . . . . . 8 |- ( ( ( 1 ... N ) \ { ( I + 1 ) } ) =/= (/) -> ran ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) = { 0 } )
76	73, 74, 75	3syl 18	. . . . . . 7 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ran ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) = { 0 } )
77	55, 76	uneq12d 3768	. . . . . 6 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ( ran { <. ( I + 1 ) , 1 >. } u. ran ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) = ( { 1 } u. { 0 } ) )
78		rnun 5541	. . . . . 6 |- ran ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) = ( ran { <. ( I + 1 ) , 1 >. } u. ran ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )
79		df-pr 4180	. . . . . 6 |- { 1 , 0 } = ( { 1 } u. { 0 } )
80	77, 78, 79	3eqtr4g 2681	. . . . 5 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ran ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) = { 1 , 0 } )
81	52, 80	eqeq12d 2637	. . . 4 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ( ran ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) = ran ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) <-> { -u 1 , 0 } = { 1 , 0 } ) )
82	25, 81	mtbiri 317	. . 3 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> -. ran ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) = ran ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) )
83		rneq 5351	. . 3 |- ( ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) -> ran ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) = ran ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) )
84	82, 83	nsyl 135	. 2 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> -. ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) )
85		axlowdimlem13.1	. . . 4 |- P = ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) )
86		axlowdimlem13.2	. . . 4 |- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )
87	85, 86	eqeq12i 2636	. . 3 |- ( P = Q <-> ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) )
88	87	necon3abii 2840	. 2 |- ( P =/= Q <-> -. ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) ) )
89	84, 88	sylibr 224	1 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> P =/= Q )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   X. cxp 5112   ran crn 5115   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   3c3 11071   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  axlowdimlem15  25836