Theorem slotsdifunifndx 13335

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 9215	. . . . 5 |- 2 e. RR
2		1nn 9156	. . . . . 6 |- 1 e. NN
3		3nn0 9422	. . . . . 6 |- 3 e. NN0
4		2nn0 9421	. . . . . 6 |- 2 e. NN0
5		2lt10 9750	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9650	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8278	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 13212	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 13329	. . . . 5 |- ( UnifSet ` ndx ) = ; 1 3
10	8, 9	neeq12i 2418	. . . 4 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) <-> 2 =/= ; 1 3 )
11	7, 10	mpbir 146	. . 3 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
12		3re 9219	. . . . 5 |- 3 e. RR
13		3lt10 9749	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9650	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8278	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 13233	. . . . 5 |- ( .r ` ndx ) = 3
17	16, 9	neeq12i 2418	. . . 4 |- ( ( .r ` ndx ) =/= ( UnifSet ` ndx ) <-> 3 =/= ; 1 3 )
18	15, 17	mpbir 146	. . 3 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
19		4re 9222	. . . . 5 |- 4 e. RR
20		4nn0 9423	. . . . . 6 |- 4 e. NN0
21		4lt10 9748	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9650	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8278	. . . 4 |- 4 =/= ; 1 3
24		starvndx 13242	. . . . 5 |- ( *r ` ndx ) = 4
25	24, 9	neeq12i 2418	. . . 4 |- ( ( *r ` ndx ) =/= ( UnifSet ` ndx ) <-> 4 =/= ; 1 3 )
26	23, 25	mpbir 146	. . 3 |- ( *r ` ndx ) =/= ( UnifSet ` ndx )
27	11, 18, 26	3pm3.2i 1201	. 2 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) )
28		10re 9631	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9420	. . . . . 6 |- 1 e. NN0
30		0nn0 9419	. . . . . 6 |- 0 e. NN0
31		3nn 9308	. . . . . 6 |- 3 e. NN
32		3pos 9239	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9640	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8278	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 13303	. . . . 5 |- ( le ` ndx ) = ; 1 0
36	35, 9	neeq12i 2418	. . . 4 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 0 =/= ; 1 3 )
37	34, 36	mpbir 146	. . 3 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
38		2nn 9307	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9632	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 9154	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9316	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9640	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8278	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 13318	. . . . 5 |- ( dist ` ndx ) = ; 1 2
45	44, 9	neeq12i 2418	. . . 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 1004   =/= wne 2401   ` cfv 5325   0cc0 8034   1c1 8035   2c2 9196   3c3 9197   4c4 9198  ;cdc 9613   ndxcnx 13099   +g cplusg 13180   .rcmulr 13181   *rcstv 13182   lecple 13187   distcds 13189   UnifSetcunif 13190

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-mulrcl 8133  ax-addcom 8134  ax-mulcom 8135  ax-addass 8136  ax-mulass 8137  ax-distr 8138  ax-i2m1 8139  ax-0lt1 8140  ax-1rid 8141  ax-0id 8142  ax-rnegex 8143  ax-precex 8144  ax-cnre 8145  ax-pre-ltirr 8146  ax-pre-ltwlin 8147  ax-pre-lttrn 8148  ax-pre-ltadd 8150  ax-pre-mulgt0 8151

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-br 4088  df-opab 4150  df-mpt 4151  df-id 4389  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-iota 5285  df-fun 5327  df-fv 5333  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-pnf 8218  df-mnf 8219  df-xr 8220  df-ltxr 8221  df-le 8222  df-sub 8354  df-neg 8355  df-inn 9146  df-2 9204  df-3 9205  df-4 9206  df-5 9207  df-6 9208  df-7 9209  df-8 9210  df-9 9211  df-n0 9405  df-z 9482  df-dec 9614  df-ndx 13105  df-slot 13106  df-plusg 13193  df-mulr 13194  df-starv 13195  df-ple 13200  df-ds 13202  df-unif 13203

This theorem is referenced by: (None)