Theorem slotsdifunifndx 13395

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 9272	. . . . 5 |- 2 e. RR
2		1nn 9213	. . . . . 6 |- 1 e. NN
3		3nn0 9479	. . . . . 6 |- 3 e. NN0
4		2nn0 9478	. . . . . 6 |- 2 e. NN0
5		2lt10 9809	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9709	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8335	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 13272	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 13389	. . . . 5 |- ( UnifSet ` ndx ) = ; 1 3
10	8, 9	neeq12i 2420	. . . 4 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) <-> 2 =/= ; 1 3 )
11	7, 10	mpbir 146	. . 3 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
12		3re 9276	. . . . 5 |- 3 e. RR
13		3lt10 9808	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9709	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8335	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 13293	. . . . 5 |- ( .r ` ndx ) = 3
17	16, 9	neeq12i 2420	. . . 4 |- ( ( .r ` ndx ) =/= ( UnifSet ` ndx ) <-> 3 =/= ; 1 3 )
18	15, 17	mpbir 146	. . 3 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
19		4re 9279	. . . . 5 |- 4 e. RR
20		4nn0 9480	. . . . . 6 |- 4 e. NN0
21		4lt10 9807	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9709	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8335	. . . 4 |- 4 =/= ; 1 3
24		starvndx 13302	. . . . 5 |- ( *r ` ndx ) = 4
25	24, 9	neeq12i 2420	. . . 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 9690	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9477	. . . . . 6 |- 1 e. NN0
30		0nn0 9476	. . . . . 6 |- 0 e. NN0
31		3nn 9365	. . . . . 6 |- 3 e. NN
32		3pos 9296	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9699	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8335	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 13363	. . . . 5 |- ( le ` ndx ) = ; 1 0
36	35, 9	neeq12i 2420	. . . 4 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 0 =/= ; 1 3 )
37	34, 36	mpbir 146	. . 3 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
38		2nn 9364	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9691	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 9211	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9373	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9699	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8335	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 13378	. . . . 5 |- ( dist ` ndx ) = ; 1 2
45	44, 9	neeq12i 2420	. . . 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 2403   ` cfv 5333   0cc0 8092   1c1 8093   2c2 9253   3c3 9254   4c4 9255  ;cdc 9672   ndxcnx 13159   +g cplusg 13240   .rcmulr 13241   *rcstv 13242   lecple 13247   distcds 13249   UnifSetcunif 13250

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208  ax-pre-mulgt0 8209

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-9 9268  df-n0 9462  df-z 9541  df-dec 9673  df-ndx 13165  df-slot 13166  df-plusg 13253  df-mulr 13254  df-starv 13255  df-ple 13260  df-ds 13262  df-unif 13263

This theorem is referenced by: (None)