Theorem slotsdifunifndx 12905

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 9060	. . . . 5 |- 2 e. RR
2		1nn 9001	. . . . . 6 |- 1 e. NN
3		3nn0 9267	. . . . . 6 |- 3 e. NN0
4		2nn0 9266	. . . . . 6 |- 2 e. NN0
5		2lt10 9594	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9494	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8123	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 12787	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 12899	. . . . 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 9064	. . . . 5 |- 3 e. RR
13		3lt10 9593	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9494	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8123	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 12807	. . . . 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 9067	. . . . 5 |- 4 e. RR
20		4nn0 9268	. . . . . 6 |- 4 e. NN0
21		4lt10 9592	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9494	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8123	. . . 4 |- 4 =/= ; 1 3
24		starvndx 12816	. . . . 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 9475	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9265	. . . . . 6 |- 1 e. NN0
30		0nn0 9264	. . . . . 6 |- 0 e. NN0
31		3nn 9153	. . . . . 6 |- 3 e. NN
32		3pos 9084	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9484	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8123	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 12877	. . . . 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 9152	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9476	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 8999	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9161	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9484	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8123	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 12888	. . . . 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 5258   0cc0 7879   1c1 7880   2c2 9041   3c3 9042   4c4 9043  ;cdc 9457   ndxcnx 12675   +g cplusg 12755   .rcmulr 12756   *rcstv 12757   lecple 12762   distcds 12764   UnifSetcunif 12765

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995  ax-pre-mulgt0 7996

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 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-n0 9250  df-z 9327  df-dec 9458  df-ndx 12681  df-slot 12682  df-plusg 12768  df-mulr 12769  df-starv 12770  df-ple 12775  df-ds 12777  df-unif 12778

This theorem is referenced by: (None)