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Mirrors > Home > MPE Home > Th. List > hashfzp1 | Structured version Visualization version GIF version |
Description: Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.) |
Ref | Expression |
---|---|
hashfzp1 | ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hash0 13019 | . . . 4 ⊢ (#‘∅) = 0 | |
2 | eluzelre 11574 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℝ) | |
3 | 2 | ltp1d 10833 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 < (𝐵 + 1)) |
4 | eluzelz 11573 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
5 | peano2z 11295 | . . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (𝐵 + 1) ∈ ℤ) | |
6 | 5 | ancri 573 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → ((𝐵 + 1) ∈ ℤ ∧ 𝐵 ∈ ℤ)) |
7 | fzn 12228 | . . . . . . 7 ⊢ (((𝐵 + 1) ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 < (𝐵 + 1) ↔ ((𝐵 + 1)...𝐵) = ∅)) | |
8 | 4, 6, 7 | 3syl 18 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 < (𝐵 + 1) ↔ ((𝐵 + 1)...𝐵) = ∅)) |
9 | 3, 8 | mpbid 221 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐵 + 1)...𝐵) = ∅) |
10 | 9 | fveq2d 6107 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐵 + 1)...𝐵)) = (#‘∅)) |
11 | 4 | zcnd 11359 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℂ) |
12 | 11 | subidd 10259 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 − 𝐵) = 0) |
13 | 1, 10, 12 | 3eqtr4a 2670 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐵 + 1)...𝐵)) = (𝐵 − 𝐵)) |
14 | oveq1 6556 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 + 1) = (𝐵 + 1)) | |
15 | 14 | oveq1d 6564 | . . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 + 1)...𝐵) = ((𝐵 + 1)...𝐵)) |
16 | 15 | fveq2d 6107 | . . . 4 ⊢ (𝐴 = 𝐵 → (#‘((𝐴 + 1)...𝐵)) = (#‘((𝐵 + 1)...𝐵))) |
17 | oveq2 6557 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 − 𝐴) = (𝐵 − 𝐵)) | |
18 | 16, 17 | eqeq12d 2625 | . . 3 ⊢ (𝐴 = 𝐵 → ((#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴) ↔ (#‘((𝐵 + 1)...𝐵)) = (𝐵 − 𝐵))) |
19 | 13, 18 | syl5ibr 235 | . 2 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴))) |
20 | uzp1 11597 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 = 𝐴 ∨ 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) | |
21 | pm2.24 120 | . . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) | |
22 | 21 | eqcoms 2618 | . . . . . . . 8 ⊢ (𝐵 = 𝐴 → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) |
23 | ax-1 6 | . . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘(𝐴 + 1)) → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) | |
24 | 22, 23 | jaoi 393 | . . . . . . 7 ⊢ ((𝐵 = 𝐴 ∨ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) |
25 | 20, 24 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) |
26 | 25 | impcom 445 | . . . . 5 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) |
27 | hashfz 13074 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘(𝐴 + 1)) → (#‘((𝐴 + 1)...𝐵)) = ((𝐵 − (𝐴 + 1)) + 1)) | |
28 | 26, 27 | syl 17 | . . . 4 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (#‘((𝐴 + 1)...𝐵)) = ((𝐵 − (𝐴 + 1)) + 1)) |
29 | eluzel2 11568 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℤ) | |
30 | 29 | zcnd 11359 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℂ) |
31 | 1cnd 9935 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 1 ∈ ℂ) | |
32 | 11, 30, 31 | nppcan2d 10297 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐵 − (𝐴 + 1)) + 1) = (𝐵 − 𝐴)) |
33 | 32 | adantl 481 | . . . 4 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((𝐵 − (𝐴 + 1)) + 1) = (𝐵 − 𝐴)) |
34 | 28, 33 | eqtrd 2644 | . . 3 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴)) |
35 | 34 | ex 449 | . 2 ⊢ (¬ 𝐴 = 𝐵 → (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴))) |
36 | 19, 35 | pm2.61i 175 | 1 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 − cmin 10145 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: 2lgslem1 24919 |
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