Theorem slotsdifunifndx 13281

Description: The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.)

Assertion

Ref	Expression
slotsdifunifndx	|- ( ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) ) /\ ( ( le ` ndx ) =/= ( UnifSet ` ndx ) /\ ( dist ` ndx ) =/= ( UnifSet ` ndx ) ) )

Proof of Theorem slotsdifunifndx

Step	Hyp	Ref	Expression
1		2re 9191	. . . . 5 |- 2 e. RR
2		1nn 9132	. . . . . 6 |- 1 e. NN
3		3nn0 9398	. . . . . 6 |- 3 e. NN0
4		2nn0 9397	. . . . . 6 |- 2 e. NN0
5		2lt10 9726	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9626	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8254	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 13158	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 13275	. . . . 5 |- ( UnifSet ` ndx ) = ; 1 3
10	8, 9	neeq12i 2417	. . . 4 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) <-> 2 =/= ; 1 3 )
11	7, 10	mpbir 146	. . 3 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
12		3re 9195	. . . . 5 |- 3 e. RR
13		3lt10 9725	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9626	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8254	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 13179	. . . . 5 |- ( .r ` ndx ) = 3
17	16, 9	neeq12i 2417	. . . 4 |- ( ( .r ` ndx ) =/= ( UnifSet ` ndx ) <-> 3 =/= ; 1 3 )
18	15, 17	mpbir 146	. . 3 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
19		4re 9198	. . . . 5 |- 4 e. RR
20		4nn0 9399	. . . . . 6 |- 4 e. NN0
21		4lt10 9724	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9626	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8254	. . . 4 |- 4 =/= ; 1 3
24		starvndx 13188	. . . . 5 |- ( *r ` ndx ) = 4
25	24, 9	neeq12i 2417	. . . 4 |- ( ( *r ` ndx ) =/= ( UnifSet ` ndx ) <-> 4 =/= ; 1 3 )
26	23, 25	mpbir 146	. . 3 |- ( *r ` ndx ) =/= ( UnifSet ` ndx )
27	11, 18, 26	3pm3.2i 1199	. 2 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) )
28		10re 9607	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9396	. . . . . 6 |- 1 e. NN0
30		0nn0 9395	. . . . . 6 |- 0 e. NN0
31		3nn 9284	. . . . . 6 |- 3 e. NN
32		3pos 9215	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9616	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8254	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 13249	. . . . 5 |- ( le ` ndx ) = ; 1 0
36	35, 9	neeq12i 2417	. . . 4 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 0 =/= ; 1 3 )
37	34, 36	mpbir 146	. . 3 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
38		2nn 9283	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9608	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 9130	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9292	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9616	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8254	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 13264	. . . . 5 |- ( dist ` ndx ) = ; 1 2
45	44, 9	neeq12i 2417	. . . 4 |- ( ( dist ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 2 =/= ; 1 3 )
46	43, 45	mpbir 146	. . 3 |- ( dist ` ndx ) =/= ( UnifSet ` ndx )
47	37, 46	pm3.2i 272	. 2 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) /\ ( dist ` ndx ) =/= ( UnifSet ` ndx ) )
48	27, 47	pm3.2i 272	1 |- ( ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) ) /\ ( ( le ` ndx ) =/= ( UnifSet ` ndx ) /\ ( dist ` ndx ) =/= ( UnifSet ` ndx ) ) )

Syntax hints:   /\ wa 104   /\ w3a 1002   =/= wne 2400   ` cfv 5318   0cc0 8010   1c1 8011   2c2 9172   3c3 9173   4c4 9174  ;cdc 9589   ndxcnx 13045   +g cplusg 13126   .rcmulr 13127   *rcstv 13128   lecple 13133   distcds 13135   UnifSetcunif 13136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-ltadd 8126  ax-pre-mulgt0 8127

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-z 9458  df-dec 9590  df-ndx 13051  df-slot 13052  df-plusg 13139  df-mulr 13140  df-starv 13141  df-ple 13146  df-ds 13148  df-unif 13149

This theorem is referenced by: (None)