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Theorem clwwlknonex2lem1 16287
Description: Lemma 1 for clwwlknonex2 16289: 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 9766 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
2 2cnd 9215 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC )
31, 2subcld 8489 . . . . . 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 9215 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> 2 e. CC )
9 1cnd 8194 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> 1 e. CC )
107, 8, 9addsubd 8510 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( (♯ ` W ) + 2 ) - 1 ) = ( ( (♯ ` W ) - 1 ) + 2 ) )
1110oveq2d 6033 . 2 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( 0..^ ( ( (♯ ` W ) - 1 ) + 2 ) ) )
12 oveq1 6024 . . . . 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 9787 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 3 ) e. NN0 )
15 1cnd 8194 . . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> 1 e. CC )
161, 2, 15subsub4d 8520 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - ( 2 + 1 ) ) )
17 2p1e3 9276 . . . . . . . 8 |- ( 2 + 1 ) = 3
1817oveq2i 6028 . . . . . . 7 |- ( N - ( 2 + 1 ) ) = ( N - 3 )
1916, 18eqtrdi 2280 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - 3 ) )
20 nn0uz 9790 . . . . . . . 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 10478 . . 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 8493 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( (♯ ` W ) - 1 ) + 1 ) = (♯ ` W ) )
2928preq2d 3755 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> { ( (♯ ` W ) - 1 ) , ( ( (♯ ` W ) - 1 ) + 1 ) } = { ( (♯ ` W ) - 1 ) , (♯ ` W ) } )
3029uneq2d 3361 . 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 1397   e. wcel 2202   u. cun 3198   {cpr 3670   ` cfv 5326  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   + caddc 8034   - cmin 8349   2c2 9193   3c3 9194   NN0cn0 9401   ZZ>=cuz 9754  ..^cfzo 10376  ♯chash 11036
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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147
This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-3 9202  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377
This theorem is referenced by:  clwwlknonex2  16289
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