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Theorem clwwlknonex2lem1 16678
Description: Lemma 1 for clwwlknonex2 16680: 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 9933 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
2 2cnd 9377 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC )
31, 2subcld 8637 . . . . . 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 2301 . . . . . 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 9377 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> 2 e. CC )
9 1cnd 8342 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> 1 e. CC )
107, 8, 9addsubd 8658 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( (♯ ` W ) + 2 ) - 1 ) = ( ( (♯ ` W ) - 1 ) + 2 ) )
1110oveq2d 6101 . 2 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( 0..^ ( ( (♯ ` W ) - 1 ) + 2 ) ) )
12 oveq1 6092 . . . . 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 9954 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 3 ) e. NN0 )
15 1cnd 8342 . . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> 1 e. CC )
161, 2, 15subsub4d 8668 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - ( 2 + 1 ) ) )
17 2p1e3 9438 . . . . . . . 8 |- ( 2 + 1 ) = 3
1817oveq2i 6096 . . . . . . 7 |- ( N - ( 2 + 1 ) ) = ( N - 3 )
1916, 18eqtrdi 2287 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - 3 ) )
20 nn0uz 9957 . . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
2120eqcomi 2242 . . . . . . 7 |- ( ZZ>= ` 0 ) = NN0
2221a1i 9 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( ZZ>= ` 0 ) = NN0 )
2314, 19, 223eltr4d 2322 . . . . 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 2315 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
26 fzosplitpr 10652 . . 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 8641 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( (♯ ` W ) - 1 ) + 1 ) = (♯ ` W ) )
2928preq2d 3795 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> { ( (♯ ` W ) - 1 ) , ( ( (♯ ` W ) - 1 ) + 1 ) } = { ( (♯ ` W ) - 1 ) , (♯ ` W ) } )
3029uneq2d 3383 . 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 2275 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   u. cun 3218   {cpr 3710   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   - cmin 8497   2c2 9355   3c3 9356   NN0cn0 9563   ZZ>=cuz 9921  ..^cfzo 10549  ♯chash 11214
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 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295
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-csb 3148  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-iun 4014  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-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-3 9364  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550
This theorem is used by:  clwwlknonex2  16680
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