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Theorem clwwlknonex2lem1 16361
Description: Lemma 1 for clwwlknonex2 16363: 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 9811 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
2 2cnd 9258 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC )
31, 2subcld 8532 . . . . . 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 2294 . . . . . 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 9258 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> 2 e. CC )
9 1cnd 8238 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> 1 e. CC )
107, 8, 9addsubd 8553 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( (♯ ` W ) + 2 ) - 1 ) = ( ( (♯ ` W ) - 1 ) + 2 ) )
1110oveq2d 6044 . 2 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( 0..^ ( ( (♯ ` W ) - 1 ) + 2 ) ) )
12 oveq1 6035 . . . . 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 9832 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 3 ) e. NN0 )
15 1cnd 8238 . . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> 1 e. CC )
161, 2, 15subsub4d 8563 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - ( 2 + 1 ) ) )
17 2p1e3 9319 . . . . . . . 8 |- ( 2 + 1 ) = 3
1817oveq2i 6039 . . . . . . 7 |- ( N - ( 2 + 1 ) ) = ( N - 3 )
1916, 18eqtrdi 2280 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - 3 ) )
20 nn0uz 9835 . . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
2120eqcomi 2235 . . . . . . 7 |- ( ZZ>= ` 0 ) = NN0
2221a1i 9 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( ZZ>= ` 0 ) = NN0 )
2314, 19, 223eltr4d 2315 . . . . 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 2308 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
26 fzosplitpr 10525 . . 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 8536 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( (♯ ` W ) - 1 ) + 1 ) = (♯ ` W ) )
2928preq2d 3759 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> { ( (♯ ` W ) - 1 ) , ( ( (♯ ` W ) - 1 ) + 1 ) } = { ( (♯ ` W ) - 1 ) , (♯ ` W ) } )
3029uneq2d 3363 . 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 2268 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 1398   e. wcel 2202   u. cun 3199   {cpr 3674   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   - cmin 8392   2c2 9236   3c3 9237   NN0cn0 9444   ZZ>=cuz 9799  ..^cfzo 10422  ♯chash 11083
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 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191
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-csb 3129  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-iun 3977  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-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-2 9244  df-3 9245  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289  df-fzo 10423
This theorem is referenced by:  clwwlknonex2  16363
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