Theorem slotsdifunifndx 13639

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 9377	. . . . 5 |- 2 e. RR
2		1nn 9318	. . . . . 6 |- 1 e. NN
3		3nn0 9586	. . . . . 6 |- 3 e. NN0
4		2nn0 9585	. . . . . 6 |- 2 e. NN0
5		2lt10 9924	. . . . . 6 |- 2 < ; 1 0
6	2, 3, 4, 5	declti 9824	. . . . 5 |- 2 < ; 1 3
7	1, 6	ltneii 8424	. . . 4 |- 2 =/= ; 1 3
8		plusgndx 13516	. . . . 5 |- ( +g ` ndx ) = 2
9		unifndx 13633	. . . . 5 |- ( UnifSet ` ndx ) = ; 1 3
10	8, 9	neeq12i 2437	. . . 4 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) <-> 2 =/= ; 1 3 )
11	7, 10	mpbir 146	. . 3 |- ( +g ` ndx ) =/= ( UnifSet ` ndx )
12		3re 9381	. . . . 5 |- 3 e. RR
13		3lt10 9923	. . . . . 6 |- 3 < ; 1 0
14	2, 3, 3, 13	declti 9824	. . . . 5 |- 3 < ; 1 3
15	12, 14	ltneii 8424	. . . 4 |- 3 =/= ; 1 3
16		mulrndx 13537	. . . . 5 |- ( .r ` ndx ) = 3
17	16, 9	neeq12i 2437	. . . 4 |- ( ( .r ` ndx ) =/= ( UnifSet ` ndx ) <-> 3 =/= ; 1 3 )
18	15, 17	mpbir 146	. . 3 |- ( .r ` ndx ) =/= ( UnifSet ` ndx )
19		4re 9384	. . . . 5 |- 4 e. RR
20		4nn0 9587	. . . . . 6 |- 4 e. NN0
21		4lt10 9922	. . . . . 6 |- 4 < ; 1 0
22	2, 3, 20, 21	declti 9824	. . . . 5 |- 4 < ; 1 3
23	19, 22	ltneii 8424	. . . 4 |- 4 =/= ; 1 3
24		starvndx 13546	. . . . 5 |- ( *r ` ndx ) = 4
25	24, 9	neeq12i 2437	. . . 4 |- ( ( *r ` ndx ) =/= ( UnifSet ` ndx ) <-> 4 =/= ; 1 3 )
26	23, 25	mpbir 146	. . 3 |- ( *r ` ndx ) =/= ( UnifSet ` ndx )
27	11, 18, 26	3pm3.2i 1206	. 2 |- ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) )
28		10re 9804	. . . . 5 |- ; 1 0 e. RR
29		1nn0 9584	. . . . . 6 |- 1 e. NN0
30		0nn0 9583	. . . . . 6 |- 0 e. NN0
31		3nn 9472	. . . . . 6 |- 3 e. NN
32		3pos 9401	. . . . . 6 |- 0 < 3
33	29, 30, 31, 32	declt 9814	. . . . 5 |- ; 1 0 < ; 1 3
34	28, 33	ltneii 8424	. . . 4 |- ; 1 0 =/= ; 1 3
35		plendx 13607	. . . . 5 |- ( le ` ndx ) = ; 1 0
36	35, 9	neeq12i 2437	. . . 4 |- ( ( le ` ndx ) =/= ( UnifSet ` ndx ) <-> ; 1 0 =/= ; 1 3 )
37	34, 36	mpbir 146	. . 3 |- ( le ` ndx ) =/= ( UnifSet ` ndx )
38		2nn 9471	. . . . . . 7 |- 2 e. NN
39	29, 38	decnncl 9805	. . . . . 6 |- ; 1 2 e. NN
40	39	nnrei 9316	. . . . 5 |- ; 1 2 e. RR
41		2lt3 9480	. . . . . 6 |- 2 < 3
42	29, 4, 31, 41	declt 9814	. . . . 5 |- ; 1 2 < ; 1 3
43	40, 42	ltneii 8424	. . . 4 |- ; 1 2 =/= ; 1 3
44		dsndx 13622	. . . . 5 |- ( dist ` ndx ) = ; 1 2
45	44, 9	neeq12i 2437	. . . 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 ) ) )

This proof depends on syntax axioms:   /\ wa 104   /\ w3a 1009   =/= wne 2420   ` cfv 5377   0cc0 8180   1c1 8181   2c2 9358   3c3 9359   4c4 9360  ;cdc 9782   ndxcnx 13401   +g cplusg 13484   .rcmulr 13485   *rcstv 13486   lecple 13491   distcds 13493   UnifSetcunif 13494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-ltadd 8296  ax-pre-mulgt0 8297

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-z 9650  df-dec 9783  df-ndx 13407  df-slot 13408  df-plusg 13497  df-mulr 13498  df-starv 13499  df-ple 13504  df-ds 13506  df-unif 13507

This theorem is used by: (None)