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Theorem clwwlknonex2lem1 16592
Description: Lemma 1 for clwwlknonex2 16594: 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 9912 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
2 2cnd 9356 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC )
31, 2subcld 8627 . . . . . 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 9356 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> 2 e. CC )
9 1cnd 8332 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> 1 e. CC )
107, 8, 9addsubd 8648 . . 3 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( (♯ ` W ) + 2 ) - 1 ) = ( ( (♯ ` W ) - 1 ) + 2 ) )
1110oveq2d 6091 . 2 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( 0..^ ( ( (♯ ` W ) - 1 ) + 2 ) ) )
12 oveq1 6082 . . . . 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 9933 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 3 ) e. NN0 )
15 1cnd 8332 . . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> 1 e. CC )
161, 2, 15subsub4d 8658 . . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - ( 2 + 1 ) ) )
17 2p1e3 9417 . . . . . . . 8 |- ( 2 + 1 ) = 3
1817oveq2i 6086 . . . . . . 7 |- ( N - ( 2 + 1 ) ) = ( N - 3 )
1916, 18eqtrdi 2287 . . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - 3 ) )
20 nn0uz 9936 . . . . . . . 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 10630 . . 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 8631 . . . 4 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( ( (♯ ` W ) - 1 ) + 1 ) = (♯ ` W ) )
2928preq2d 3791 . . 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
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   u. cun 3218   {cpr 3706   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   - cmin 8487   2c2 9334   3c3 9335   NN0cn0 9542   ZZ>=cuz 9900  ..^cfzo 10527  ♯chash 11192
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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285
This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-3 9343  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528
This theorem is referenced by:  clwwlknonex2  16594
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