Theorem slotsdifunifndx 13445

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 9307	. . . . 5 |- 2 e. RR
2		1nn 9248	. . . . . 6 |- 1 e. NN
3		3nn0 9514	. . . . . 6 |- 3 e. NN0
4		2nn0 9513	. . . . . 6 |- 2 e. NN0
5		2lt10 9846	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9746	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8370	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 13322	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 13439	. . . . 5 |- ( UnifSet ` ndx ) = ; 1 3
10	8, 9	neeq12i 2429	. . . 4 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) <-> 2 =/= ; 1 3 )
11	7, 10	mpbir 146	. . 3 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
12		3re 9311	. . . . 5 |- 3 e. RR
13		3lt10 9845	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9746	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8370	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 13343	. . . . 5 |- ( .r ` ndx ) = 3
17	16, 9	neeq12i 2429	. . . 4 |- ( ( .r ` ndx ) =/= ( UnifSet ` ndx ) <-> 3 =/= ; 1 3 )
18	15, 17	mpbir 146	. . 3 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
19		4re 9314	. . . . 5 |- 4 e. RR
20		4nn0 9515	. . . . . 6 |- 4 e. NN0
21		4lt10 9844	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9746	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8370	. . . 4 |- 4 =/= ; 1 3
24		starvndx 13352	. . . . 5 |- ( *r ` ndx ) = 4
25	24, 9	neeq12i 2429	. . . 4 |- ( ( *r ` ndx ) =/= ( UnifSet ` ndx ) <-> 4 =/= ; 1 3 )
26	23, 25	mpbir 146	. . 3 |- ( *r ` ndx ) =/= ( UnifSet ` ndx )
27	11, 18, 26	3pm3.2i 1202	. 2 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) )
28		10re 9727	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9512	. . . . . 6 |- 1 e. NN0
30		0nn0 9511	. . . . . 6 |- 0 e. NN0
31		3nn 9400	. . . . . 6 |- 3 e. NN
32		3pos 9331	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9736	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8370	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 13413	. . . . 5 |- ( le ` ndx ) = ; 1 0
36	35, 9	neeq12i 2429	. . . 4 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 0 =/= ; 1 3 )
37	34, 36	mpbir 146	. . 3 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
38		2nn 9399	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9728	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 9246	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9408	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9736	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8370	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 13428	. . . . 5 |- ( dist ` ndx ) = ; 1 2
45	44, 9	neeq12i 2429	. . . 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 1005   =/= wne 2412   ` cfv 5352   0cc0 8127   1c1 8128   2c2 9288   3c3 9289   4c4 9290  ;cdc 9709   ndxcnx 13209   +g cplusg 13290   .rcmulr 13291   *rcstv 13292   lecple 13297   distcds 13299   UnifSetcunif 13300

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-z 9578  df-dec 9710  df-ndx 13215  df-slot 13216  df-plusg 13303  df-mulr 13304  df-starv 13305  df-ple 13310  df-ds 13312  df-unif 13313

This theorem is referenced by: (None)