Theorem slotsdifunifndx 13314

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 9212	. . . . 5 |- 2 e. RR
2		1nn 9153	. . . . . 6 |- 1 e. NN
3		3nn0 9419	. . . . . 6 |- 3 e. NN0
4		2nn0 9418	. . . . . 6 |- 2 e. NN0
5		2lt10 9747	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9647	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8275	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 13191	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 13308	. . . . 5 |- ( UnifSet ` ndx ) = ; 1 3
10	8, 9	neeq12i 2419	. . . 4 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) <-> 2 =/= ; 1 3 )
11	7, 10	mpbir 146	. . 3 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
12		3re 9216	. . . . 5 |- 3 e. RR
13		3lt10 9746	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9647	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8275	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 13212	. . . . 5 |- ( .r ` ndx ) = 3
17	16, 9	neeq12i 2419	. . . 4 |- ( ( .r ` ndx ) =/= ( UnifSet ` ndx ) <-> 3 =/= ; 1 3 )
18	15, 17	mpbir 146	. . 3 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
19		4re 9219	. . . . 5 |- 4 e. RR
20		4nn0 9420	. . . . . 6 |- 4 e. NN0
21		4lt10 9745	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9647	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8275	. . . 4 |- 4 =/= ; 1 3
24		starvndx 13221	. . . . 5 |- ( *r ` ndx ) = 4
25	24, 9	neeq12i 2419	. . . 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 9628	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9417	. . . . . 6 |- 1 e. NN0
30		0nn0 9416	. . . . . 6 |- 0 e. NN0
31		3nn 9305	. . . . . 6 |- 3 e. NN
32		3pos 9236	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9637	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8275	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 13282	. . . . 5 |- ( le ` ndx ) = ; 1 0
36	35, 9	neeq12i 2419	. . . 4 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 0 =/= ; 1 3 )
37	34, 36	mpbir 146	. . 3 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
38		2nn 9304	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9629	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 9151	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9313	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9637	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8275	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 13297	. . . . 5 |- ( dist ` ndx ) = ; 1 2
45	44, 9	neeq12i 2419	. . . 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 2402   ` cfv 5326   0cc0 8031   1c1 8032   2c2 9193   3c3 9194   4c4 9195  ;cdc 9610   ndxcnx 13078   +g cplusg 13159   .rcmulr 13160   *rcstv 13161   lecple 13166   distcds 13168   UnifSetcunif 13169

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-dec 9611  df-ndx 13084  df-slot 13085  df-plusg 13172  df-mulr 13173  df-starv 13174  df-ple 13179  df-ds 13181  df-unif 13182

This theorem is referenced by: (None)