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Mirrors > Home > MPE Home > Th. List > clwwlknonex2lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for clwwlknonex2 27880: 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} ≠ {-1, 0} = (∅ ∪ {-1, 0}) = ((0..^(0 − 1)) ∪ {(0 − 1), 0}) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (♯‘𝑊)}). (Contributed by AV, 22-Sep-2018.) (Revised by AV, 26-Jan-2022.) |
Ref | Expression |
---|---|
clwwlknonex2lem1 | ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → (0..^(((♯‘𝑊) + 2) − 1)) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (♯‘𝑊)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelcn 12247 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℂ) | |
2 | 2cnd 11707 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ ℂ) | |
3 | 1, 2 | subcld 10989 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℂ) |
4 | 3 | adantr 483 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → (𝑁 − 2) ∈ ℂ) |
5 | eleq1 2898 | . . . . . 6 ⊢ ((♯‘𝑊) = (𝑁 − 2) → ((♯‘𝑊) ∈ ℂ ↔ (𝑁 − 2) ∈ ℂ)) | |
6 | 5 | adantl 484 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → ((♯‘𝑊) ∈ ℂ ↔ (𝑁 − 2) ∈ ℂ)) |
7 | 4, 6 | mpbird 259 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → (♯‘𝑊) ∈ ℂ) |
8 | 2cnd 11707 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → 2 ∈ ℂ) | |
9 | 1cnd 10628 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → 1 ∈ ℂ) | |
10 | 7, 8, 9 | addsubd 11010 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → (((♯‘𝑊) + 2) − 1) = (((♯‘𝑊) − 1) + 2)) |
11 | 10 | oveq2d 7164 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → (0..^(((♯‘𝑊) + 2) − 1)) = (0..^(((♯‘𝑊) − 1) + 2))) |
12 | oveq1 7155 | . . . . 5 ⊢ ((♯‘𝑊) = (𝑁 − 2) → ((♯‘𝑊) − 1) = ((𝑁 − 2) − 1)) | |
13 | 12 | adantl 484 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → ((♯‘𝑊) − 1) = ((𝑁 − 2) − 1)) |
14 | uznn0sub 12269 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 3) ∈ ℕ0) | |
15 | 1cnd 10628 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → 1 ∈ ℂ) | |
16 | 1, 2, 15 | subsub4d 11020 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → ((𝑁 − 2) − 1) = (𝑁 − (2 + 1))) |
17 | 2p1e3 11771 | . . . . . . . 8 ⊢ (2 + 1) = 3 | |
18 | 17 | oveq2i 7159 | . . . . . . 7 ⊢ (𝑁 − (2 + 1)) = (𝑁 − 3) |
19 | 16, 18 | syl6eq 2870 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → ((𝑁 − 2) − 1) = (𝑁 − 3)) |
20 | nn0uz 12272 | . . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0) | |
21 | 20 | eqcomi 2828 | . . . . . . 7 ⊢ (ℤ≥‘0) = ℕ0 |
22 | 21 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → (ℤ≥‘0) = ℕ0) |
23 | 14, 19, 22 | 3eltr4d 2926 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘3) → ((𝑁 − 2) − 1) ∈ (ℤ≥‘0)) |
24 | 23 | adantr 483 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → ((𝑁 − 2) − 1) ∈ (ℤ≥‘0)) |
25 | 13, 24 | eqeltrd 2911 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → ((♯‘𝑊) − 1) ∈ (ℤ≥‘0)) |
26 | fzosplitpr 13138 | . . 3 ⊢ (((♯‘𝑊) − 1) ∈ (ℤ≥‘0) → (0..^(((♯‘𝑊) − 1) + 2)) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (((♯‘𝑊) − 1) + 1)})) | |
27 | 25, 26 | syl 17 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → (0..^(((♯‘𝑊) − 1) + 2)) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (((♯‘𝑊) − 1) + 1)})) |
28 | 7, 9 | npcand 10993 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → (((♯‘𝑊) − 1) + 1) = (♯‘𝑊)) |
29 | 28 | preq2d 4668 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → {((♯‘𝑊) − 1), (((♯‘𝑊) − 1) + 1)} = {((♯‘𝑊) − 1), (♯‘𝑊)}) |
30 | 29 | uneq2d 4137 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (((♯‘𝑊) − 1) + 1)}) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (♯‘𝑊)})) |
31 | 11, 27, 30 | 3eqtrd 2858 | 1 ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → (0..^(((♯‘𝑊) + 2) − 1)) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (♯‘𝑊)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∪ cun 3932 {cpr 4561 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 0cc0 10529 1c1 10530 + caddc 10532 − cmin 10862 2c2 11684 3c3 11685 ℕ0cn0 11889 ℤ≥cuz 12235 ..^cfzo 13025 ♯chash 13682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 |
This theorem is referenced by: clwwlknonex2 27880 |
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