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Theorem clwwlknonex2lem1 16222
Description: Lemma 1 for clwwlknonex2 16224: Transformation of a special half-open integer range into a union of a smaller half-open integer range and an unordered pair. This Lemma would not hold for N = 2, i.e., (♯ ` W ) = 0, because ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( 0..^ ( ( 0 + 2 ) - 1 ) ) = ( 0..^ 1 ) = { 0 } =/= { -u 1 , 0 } = ( (/) u. { -u 1 , 0 } ) = ( ( 0..^ ( 0 - 1 ) ) u. { ( 0 - 1 ) , 0 } ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ). (Contributed by AV, 22-Sep-2018.) (Revised by AV, 26-Jan-2022.)
Assertion
Ref Expression
clwwlknonex2lem1 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) )
Proof of Theorem clwwlknonex2lem1
StepHypRef Expression
1 eluzelcn 9755 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
2 2cnd 9204 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC )
31, 2subcld 8478 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. CC )
43adantr 276 . . . . 5 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( N - 2 ) e. CC )
5 eleq1 2292 . . . . . 6 |- ( (♯ ` W ) = ( N - 2 ) -> ( (♯ ` W ) e. CC <-> ( N - 2 ) e. CC ) )
65adantl 277 . . . . 5 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( (♯ ` W ) e. CC <-> ( N - 2 ) e. CC ) )
74, 6mpbird 167 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> (♯ ` W ) e. CC )
8 2cnd 9204 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> 2 e. CC )
9 1cnd 8183 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> 1 e. CC )
107, 8, 9addsubd 8499 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( (♯ ` W ) + 2 ) - 1 ) = ( ( (♯ ` W ) - 1 ) + 2 ) )
1110oveq2d 6027 . 2 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( 0..^ ( ( (♯ ` W ) - 1 ) + 2 ) ) )
12 oveq1 6018 . . . . 5 |- ( (♯ ` W ) = ( N - 2 ) -> ( (♯ ` W ) - 1 ) = ( ( N - 2 ) - 1 ) )
1312adantl 277 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( (♯ ` W ) - 1 ) = ( ( N - 2 ) - 1 ) )
14 uznn0sub 9776 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 3 ) e. NN0 )
15 1cnd 8183 . . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> 1 e. CC )
161, 2, 15subsub4d 8509 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - ( 2 + 1 ) ) )
17 2p1e3 9265 . . . . . . . 8 |- ( 2 + 1 ) = 3
1817oveq2i 6022 . . . . . . 7 |- ( N - ( 2 + 1 ) ) = ( N - 3 )
1916, 18eqtrdi 2278 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - 3 ) )
20 nn0uz 9779 . . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
2120eqcomi 2233 . . . . . . 7 |- ( ZZ>= ` 0 ) = NN0
2221a1i 9 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( ZZ>= ` 0 ) = NN0 )
2314, 19, 223eltr4d 2313 . . . . 5 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) e. ( ZZ>= ` 0 ) )
2423adantr 276 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( N - 2 ) - 1 ) e. ( ZZ>= ` 0 ) )
2513, 24eqeltrd 2306 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
26 fzosplitpr 10467 . . 3 |- ( ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) -> ( 0..^ ( ( (♯ ` W ) - 1 ) + 2 ) ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , ( ( (♯ ` W ) - 1 ) + 1 ) } ) )
2725, 26syl 14 . 2 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( (♯ ` W ) - 1 ) + 2 ) ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , ( ( (♯ ` W ) - 1 ) + 1 ) } ) )
287, 9npcand 8482 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( (♯ ` W ) - 1 ) + 1 ) = (♯ ` W ) )
2928preq2d 3751 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> { ( (♯ ` W ) - 1 ) , ( ( (♯ ` W ) - 1 ) + 1 ) } = { ( (♯ ` W ) - 1 ) , (♯ ` W ) } )
3029uneq2d 3359 . 2 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , ( ( (♯ ` W ) - 1 ) + 1 ) } ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) )
3111, 27, 303eqtrd 2266 1 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   u. cun 3196   {cpr 3668   ` cfv 5322  (class class class)co 6011   CCcc 8018   0cc0 8020   1c1 8021   + caddc 8023   - cmin 8338   2c2 9182   3c3 9183   NN0cn0 9390   ZZ>=cuz 9743  ..^cfzo 10365  ♯chash 11025
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 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-addass 8122  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136
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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-inn 9132  df-2 9190  df-3 9191  df-n0 9391  df-z 9468  df-uz 9744  df-fz 10232  df-fzo 10366
This theorem is referenced by:  clwwlknonex2  16224
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