Theorem slotsdifunifndx 12934

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 9077	. . . . 5 |- 2 e. RR
2		1nn 9018	. . . . . 6 |- 1 e. NN
3		3nn0 9284	. . . . . 6 |- 3 e. NN0
4		2nn0 9283	. . . . . 6 |- 2 e. NN0
5		2lt10 9611	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9511	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8140	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 12812	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 12928	. . . . 5 |- ( UnifSet ` ndx ) = ; 1 3
10	8, 9	neeq12i 2384	. . . 4 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) <-> 2 =/= ; 1 3 )
11	7, 10	mpbir 146	. . 3 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
12		3re 9081	. . . . 5 |- 3 e. RR
13		3lt10 9610	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9511	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8140	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 12832	. . . . 5 |- ( .r ` ndx ) = 3
17	16, 9	neeq12i 2384	. . . 4 |- ( ( .r ` ndx ) =/= ( UnifSet ` ndx ) <-> 3 =/= ; 1 3 )
18	15, 17	mpbir 146	. . 3 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
19		4re 9084	. . . . 5 |- 4 e. RR
20		4nn0 9285	. . . . . 6 |- 4 e. NN0
21		4lt10 9609	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9511	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8140	. . . 4 |- 4 =/= ; 1 3
24		starvndx 12841	. . . . 5 |- ( *r ` ndx ) = 4
25	24, 9	neeq12i 2384	. . . 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 9492	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9282	. . . . . 6 |- 1 e. NN0
30		0nn0 9281	. . . . . 6 |- 0 e. NN0
31		3nn 9170	. . . . . 6 |- 3 e. NN
32		3pos 9101	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9501	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8140	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 12902	. . . . 5 |- ( le ` ndx ) = ; 1 0
36	35, 9	neeq12i 2384	. . . 4 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 0 =/= ; 1 3 )
37	34, 36	mpbir 146	. . 3 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
38		2nn 9169	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9493	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 9016	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9178	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9501	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8140	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 12917	. . . . 5 |- ( dist ` ndx ) = ; 1 2
45	44, 9	neeq12i 2384	. . . 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 2367   ` cfv 5259   0cc0 7896   1c1 7897   2c2 9058   3c3 9059   4c4 9060  ;cdc 9474   ndxcnx 12700   +g cplusg 12780   .rcmulr 12781   *rcstv 12782   lecple 12787   distcds 12789   UnifSetcunif 12790

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012  ax-pre-mulgt0 8013

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-n0 9267  df-z 9344  df-dec 9475  df-ndx 12706  df-slot 12707  df-plusg 12793  df-mulr 12794  df-starv 12795  df-ple 12800  df-ds 12802  df-unif 12803

This theorem is referenced by: (None)