Theorem slotsdifunifndx 13265

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 9180	. . . . 5 |- 2 e. RR
2		1nn 9121	. . . . . 6 |- 1 e. NN
3		3nn0 9387	. . . . . 6 |- 3 e. NN0
4		2nn0 9386	. . . . . 6 |- 2 e. NN0
5		2lt10 9715	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9615	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8243	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 13142	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 13259	. . . . 5 |- ( UnifSet ` ndx ) = ; 1 3
10	8, 9	neeq12i 2417	. . . 4 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) <-> 2 =/= ; 1 3 )
11	7, 10	mpbir 146	. . 3 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
12		3re 9184	. . . . 5 |- 3 e. RR
13		3lt10 9714	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9615	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8243	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 13163	. . . . 5 |- ( .r ` ndx ) = 3
17	16, 9	neeq12i 2417	. . . 4 |- ( ( .r ` ndx ) =/= ( UnifSet ` ndx ) <-> 3 =/= ; 1 3 )
18	15, 17	mpbir 146	. . 3 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
19		4re 9187	. . . . 5 |- 4 e. RR
20		4nn0 9388	. . . . . 6 |- 4 e. NN0
21		4lt10 9713	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9615	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8243	. . . 4 |- 4 =/= ; 1 3
24		starvndx 13172	. . . . 5 |- ( *r ` ndx ) = 4
25	24, 9	neeq12i 2417	. . . 4 |- ( ( *r ` ndx ) =/= ( UnifSet ` ndx ) <-> 4 =/= ; 1 3 )
26	23, 25	mpbir 146	. . 3 |- ( *r ` ndx ) =/= ( UnifSet ` ndx )
27	11, 18, 26	3pm3.2i 1199	. 2 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) )
28		10re 9596	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9385	. . . . . 6 |- 1 e. NN0
30		0nn0 9384	. . . . . 6 |- 0 e. NN0
31		3nn 9273	. . . . . 6 |- 3 e. NN
32		3pos 9204	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9605	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8243	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 13233	. . . . 5 |- ( le ` ndx ) = ; 1 0
36	35, 9	neeq12i 2417	. . . 4 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 0 =/= ; 1 3 )
37	34, 36	mpbir 146	. . 3 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
38		2nn 9272	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9597	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 9119	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9281	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9605	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8243	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 13248	. . . . 5 |- ( dist ` ndx ) = ; 1 2
45	44, 9	neeq12i 2417	. . . 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 1002   =/= wne 2400   ` cfv 5318   0cc0 7999   1c1 8000   2c2 9161   3c3 9162   4c4 9163  ;cdc 9578   ndxcnx 13029   +g cplusg 13110   .rcmulr 13111   *rcstv 13112   lecple 13117   distcds 13119   UnifSetcunif 13120

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115  ax-pre-mulgt0 8116

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-z 9447  df-dec 9579  df-ndx 13035  df-slot 13036  df-plusg 13123  df-mulr 13124  df-starv 13125  df-ple 13130  df-ds 13132  df-unif 13133

This theorem is referenced by: (None)