Theorem axlowdimlem14 25835

Description: Lemma for axlowdim 25841. Take two possible Q from axlowdimlem10 25831. They are the same iff their distinguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)

Hypotheses

Ref	Expression
axlowdimlem14.1	|- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )
axlowdimlem14.2	|- R = ( { <. ( J + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( J + 1 ) } ) X. { 0 } ) )

Assertion

Ref	Expression
axlowdimlem14	|- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( Q = R -> I = J ) )

Proof of Theorem axlowdimlem14

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axlowdimlem14.1	. . . . . . 7 |- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )
2	1	axlowdimlem10 25831	. . . . . 6 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> Q e. ( EE ` N ) )
3		elee 25774	. . . . . . 7 |- ( N e. NN -> ( Q e. ( EE ` N ) <-> Q : ( 1 ... N ) --> RR ) )
4	3	adantr 481	. . . . . 6 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ( Q e. ( EE ` N ) <-> Q : ( 1 ... N ) --> RR ) )
5	2, 4	mpbid 222	. . . . 5 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> Q : ( 1 ... N ) --> RR )
6		ffn 6045	. . . . 5 |- ( Q : ( 1 ... N ) --> RR -> Q Fn ( 1 ... N ) )
7	5, 6	syl 17	. . . 4 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> Q Fn ( 1 ... N ) )
8		axlowdimlem14.2	. . . . . . 7 |- R = ( { <. ( J + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( J + 1 ) } ) X. { 0 } ) )
9	8	axlowdimlem10 25831	. . . . . 6 |- ( ( N e. NN /\ J e. ( 1 ... ( N - 1 ) ) ) -> R e. ( EE ` N ) )
10		elee 25774	. . . . . . 7 |- ( N e. NN -> ( R e. ( EE ` N ) <-> R : ( 1 ... N ) --> RR ) )
11	10	adantr 481	. . . . . 6 |- ( ( N e. NN /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( R e. ( EE ` N ) <-> R : ( 1 ... N ) --> RR ) )
12	9, 11	mpbid 222	. . . . 5 |- ( ( N e. NN /\ J e. ( 1 ... ( N - 1 ) ) ) -> R : ( 1 ... N ) --> RR )
13		ffn 6045	. . . . 5 |- ( R : ( 1 ... N ) --> RR -> R Fn ( 1 ... N ) )
14	12, 13	syl 17	. . . 4 |- ( ( N e. NN /\ J e. ( 1 ... ( N - 1 ) ) ) -> R Fn ( 1 ... N ) )
15		eqfnfv 6311	. . . 4 |- ( ( Q Fn ( 1 ... N ) /\ R Fn ( 1 ... N ) ) -> ( Q = R <-> A. i e. ( 1 ... N ) ( Q ` i ) = ( R ` i ) ) )
16	7, 14, 15	syl2an 494	. . 3 |- ( ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) /\ ( N e. NN /\ J e. ( 1 ... ( N - 1 ) ) ) ) -> ( Q = R <-> A. i e. ( 1 ... N ) ( Q ` i ) = ( R ` i ) ) )
17	16	3impdi 1381	. 2 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( Q = R <-> A. i e. ( 1 ... N ) ( Q ` i ) = ( R ` i ) ) )
18		fznatpl1 12395	. . . . . . . 8 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> ( I + 1 ) e. ( 1 ... N ) )
19	18	3adant3 1081	. . . . . . 7 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( I + 1 ) e. ( 1 ... N ) )
20	19	adantr 481	. . . . . 6 |- ( ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) /\ I =/= J ) -> ( I + 1 ) e. ( 1 ... N ) )
21		ax-1ne0 10005	. . . . . . . 8 |- 1 =/= 0
22	21	a1i 11	. . . . . . 7 |- ( ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) /\ I =/= J ) -> 1 =/= 0 )
23	1	axlowdimlem11 25832	. . . . . . . 8 |- ( Q ` ( I + 1 ) ) = 1
24	23	a1i 11	. . . . . . 7 |- ( ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) /\ I =/= J ) -> ( Q ` ( I + 1 ) ) = 1 )
25		elfzelz 12342	. . . . . . . . . . . . 13 |- ( I e. ( 1 ... ( N - 1 ) ) -> I e. ZZ )
26	25	zcnd 11483	. . . . . . . . . . . 12 |- ( I e. ( 1 ... ( N - 1 ) ) -> I e. CC )
27		elfzelz 12342	. . . . . . . . . . . . 13 |- ( J e. ( 1 ... ( N - 1 ) ) -> J e. ZZ )
28	27	zcnd 11483	. . . . . . . . . . . 12 |- ( J e. ( 1 ... ( N - 1 ) ) -> J e. CC )
29		ax-1cn 9994	. . . . . . . . . . . . 13 |- 1 e. CC
30		addcan2 10221	. . . . . . . . . . . . 13 |- ( ( I e. CC /\ J e. CC /\ 1 e. CC ) -> ( ( I + 1 ) = ( J + 1 ) <-> I = J ) )
31	29, 30	mp3an3 1413	. . . . . . . . . . . 12 |- ( ( I e. CC /\ J e. CC ) -> ( ( I + 1 ) = ( J + 1 ) <-> I = J ) )
32	26, 28, 31	syl2an 494	. . . . . . . . . . 11 |- ( ( I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( ( I + 1 ) = ( J + 1 ) <-> I = J ) )
33	32	3adant1 1079	. . . . . . . . . 10 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( ( I + 1 ) = ( J + 1 ) <-> I = J ) )
34	33	necon3bid 2838	. . . . . . . . 9 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( ( I + 1 ) =/= ( J + 1 ) <-> I =/= J ) )
35	34	biimpar 502	. . . . . . . 8 |- ( ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) /\ I =/= J ) -> ( I + 1 ) =/= ( J + 1 ) )
36	8	axlowdimlem12 25833	. . . . . . . 8 |- ( ( ( I + 1 ) e. ( 1 ... N ) /\ ( I + 1 ) =/= ( J + 1 ) ) -> ( R ` ( I + 1 ) ) = 0 )
37	20, 35, 36	syl2anc 693	. . . . . . 7 |- ( ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) /\ I =/= J ) -> ( R ` ( I + 1 ) ) = 0 )
38	22, 24, 37	3netr4d 2871	. . . . . 6 |- ( ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) /\ I =/= J ) -> ( Q ` ( I + 1 ) ) =/= ( R ` ( I + 1 ) ) )
39		df-ne 2795	. . . . . . . 8 |- ( ( Q ` i ) =/= ( R ` i ) <-> -. ( Q ` i ) = ( R ` i ) )
40		fveq2 6191	. . . . . . . . 9 |- ( i = ( I + 1 ) -> ( Q ` i ) = ( Q ` ( I + 1 ) ) )
41		fveq2 6191	. . . . . . . . 9 |- ( i = ( I + 1 ) -> ( R ` i ) = ( R ` ( I + 1 ) ) )
42	40, 41	neeq12d 2855	. . . . . . . 8 |- ( i = ( I + 1 ) -> ( ( Q ` i ) =/= ( R ` i ) <-> ( Q ` ( I + 1 ) ) =/= ( R ` ( I + 1 ) ) ) )
43	39, 42	syl5bbr 274	. . . . . . 7 |- ( i = ( I + 1 ) -> ( -. ( Q ` i ) = ( R ` i ) <-> ( Q ` ( I + 1 ) ) =/= ( R ` ( I + 1 ) ) ) )
44	43	rspcev 3309	. . . . . 6 |- ( ( ( I + 1 ) e. ( 1 ... N ) /\ ( Q ` ( I + 1 ) ) =/= ( R ` ( I + 1 ) ) ) -> E. i e. ( 1 ... N ) -. ( Q ` i ) = ( R ` i ) )
45	20, 38, 44	syl2anc 693	. . . . 5 |- ( ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) /\ I =/= J ) -> E. i e. ( 1 ... N ) -. ( Q ` i ) = ( R ` i ) )
46	45	ex 450	. . . 4 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( I =/= J -> E. i e. ( 1 ... N ) -. ( Q ` i ) = ( R ` i ) ) )
47		df-ne 2795	. . . 4 |- ( I =/= J <-> -. I = J )
48		rexnal 2995	. . . 4 |- ( E. i e. ( 1 ... N ) -. ( Q ` i ) = ( R ` i ) <-> -. A. i e. ( 1 ... N ) ( Q ` i ) = ( R ` i ) )
49	46, 47, 48	3imtr3g 284	. . 3 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( -. I = J -> -. A. i e. ( 1 ... N ) ( Q ` i ) = ( R ` i ) ) )
50	49	con4d 114	. 2 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( A. i e. ( 1 ... N ) ( Q ` i ) = ( R ` i ) -> I = J ) )
51	17, 50	sylbid 230	1 |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( Q = R -> I = J ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   u. cun 3572   {csn 4177   <.cop 4183   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   ...cfz 12326   EEcee 25768

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-map 7859  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-ee 25771

This theorem is referenced by:  axlowdimlem15  25836