Theorem slotsdifunifndx 12688

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 8991	. . . . 5 |- 2 e. RR
2		1nn 8932	. . . . . 6 |- 1 e. NN
3		3nn0 9196	. . . . . 6 |- 3 e. NN0
4		2nn0 9195	. . . . . 6 |- 2 e. NN0
5		2lt10 9523	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9423	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8056	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 12570	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 12682	. . . . 5 |- ( UnifSet ` ndx ) = ; 1 3
10	8, 9	neeq12i 2364	. . . 4 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) <-> 2 =/= ; 1 3 )
11	7, 10	mpbir 146	. . 3 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
12		3re 8995	. . . . 5 |- 3 e. RR
13		3lt10 9522	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9423	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8056	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 12590	. . . . 5 |- ( .r ` ndx ) = 3
17	16, 9	neeq12i 2364	. . . 4 |- ( ( .r ` ndx ) =/= ( UnifSet ` ndx ) <-> 3 =/= ; 1 3 )
18	15, 17	mpbir 146	. . 3 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
19		4re 8998	. . . . 5 |- 4 e. RR
20		4nn0 9197	. . . . . 6 |- 4 e. NN0
21		4lt10 9521	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9423	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8056	. . . 4 |- 4 =/= ; 1 3
24		starvndx 12599	. . . . 5 |- ( *r ` ndx ) = 4
25	24, 9	neeq12i 2364	. . . 4 |- ( ( *r ` ndx ) =/= ( UnifSet ` ndx ) <-> 4 =/= ; 1 3 )
26	23, 25	mpbir 146	. . 3 |- ( *r ` ndx ) =/= ( UnifSet ` ndx )
27	11, 18, 26	3pm3.2i 1175	. 2 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) )
28		10re 9404	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9194	. . . . . 6 |- 1 e. NN0
30		0nn0 9193	. . . . . 6 |- 0 e. NN0
31		3nn 9083	. . . . . 6 |- 3 e. NN
32		3pos 9015	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9413	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8056	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 12660	. . . . 5 |- ( le ` ndx ) = ; 1 0
36	35, 9	neeq12i 2364	. . . 4 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 0 =/= ; 1 3 )
37	34, 36	mpbir 146	. . 3 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
38		2nn 9082	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9405	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 8930	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9091	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9413	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8056	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 12671	. . . . 5 |- ( dist ` ndx ) = ; 1 2
45	44, 9	neeq12i 2364	. . . 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 978   =/= wne 2347   ` cfv 5218   0cc0 7813   1c1 7814   2c2 8972   3c3 8973   4c4 8974  ;cdc 9386   ndxcnx 12461   +g cplusg 12538   .rcmulr 12539   *rcstv 12540   lecple 12545   distcds 12547   UnifSetcunif 12548

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929  ax-pre-mulgt0 7930

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-7 8985  df-8 8986  df-9 8987  df-n0 9179  df-z 9256  df-dec 9387  df-ndx 12467  df-slot 12468  df-plusg 12551  df-mulr 12552  df-starv 12553  df-ple 12558  df-ds 12560  df-unif 12561

This theorem is referenced by: (None)