Theorem slotsdifunifndx 12845

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 9052	. . . . 5 |- 2 e. RR
2		1nn 8993	. . . . . 6 |- 1 e. NN
3		3nn0 9258	. . . . . 6 |- 3 e. NN0
4		2nn0 9257	. . . . . 6 |- 2 e. NN0
5		2lt10 9585	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9485	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8116	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 12727	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 12839	. . . . 5 |- ( UnifSet ` ndx ) = ; 1 3
10	8, 9	neeq12i 2381	. . . 4 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) <-> 2 =/= ; 1 3 )
11	7, 10	mpbir 146	. . 3 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
12		3re 9056	. . . . 5 |- 3 e. RR
13		3lt10 9584	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9485	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8116	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 12747	. . . . 5 |- ( .r ` ndx ) = 3
17	16, 9	neeq12i 2381	. . . 4 |- ( ( .r ` ndx ) =/= ( UnifSet ` ndx ) <-> 3 =/= ; 1 3 )
18	15, 17	mpbir 146	. . 3 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
19		4re 9059	. . . . 5 |- 4 e. RR
20		4nn0 9259	. . . . . 6 |- 4 e. NN0
21		4lt10 9583	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9485	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8116	. . . 4 |- 4 =/= ; 1 3
24		starvndx 12756	. . . . 5 |- ( *r ` ndx ) = 4
25	24, 9	neeq12i 2381	. . . 4 |- ( ( *r ` ndx ) =/= ( UnifSet ` ndx ) <-> 4 =/= ; 1 3 )
26	23, 25	mpbir 146	. . 3 |- ( *r ` ndx ) =/= ( UnifSet ` ndx )
27	11, 18, 26	3pm3.2i 1177	. 2 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) )
28		10re 9466	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9256	. . . . . 6 |- 1 e. NN0
30		0nn0 9255	. . . . . 6 |- 0 e. NN0
31		3nn 9144	. . . . . 6 |- 3 e. NN
32		3pos 9076	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9475	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8116	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 12817	. . . . 5 |- ( le ` ndx ) = ; 1 0
36	35, 9	neeq12i 2381	. . . 4 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 0 =/= ; 1 3 )
37	34, 36	mpbir 146	. . 3 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
38		2nn 9143	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9467	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 8991	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9152	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9475	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8116	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 12828	. . . . 5 |- ( dist ` ndx ) = ; 1 2
45	44, 9	neeq12i 2381	. . . 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 980   =/= wne 2364   ` cfv 5254   0cc0 7872   1c1 7873   2c2 9033   3c3 9034   4c4 9035  ;cdc 9448   ndxcnx 12615   +g cplusg 12695   .rcmulr 12696   *rcstv 12697   lecple 12702   distcds 12704   UnifSetcunif 12705

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988  ax-pre-mulgt0 7989

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-9 9048  df-n0 9241  df-z 9318  df-dec 9449  df-ndx 12621  df-slot 12622  df-plusg 12708  df-mulr 12709  df-starv 12710  df-ple 12715  df-ds 12717  df-unif 12718

This theorem is referenced by: (None)