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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 13158 | . . . 4 ⊢ (#‘∅) = 0 | |
2 | eluzelre 11698 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℝ) | |
3 | 2 | ltp1d 10954 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 < (𝐵 + 1)) |
4 | eluzelz 11697 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
5 | peano2z 11418 | . . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (𝐵 + 1) ∈ ℤ) | |
6 | 5 | ancri 575 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → ((𝐵 + 1) ∈ ℤ ∧ 𝐵 ∈ ℤ)) |
7 | fzn 12357 | . . . . . . 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 222 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐵 + 1)...𝐵) = ∅) |
10 | 9 | fveq2d 6195 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐵 + 1)...𝐵)) = (#‘∅)) |
11 | 4 | zcnd 11483 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℂ) |
12 | 11 | subidd 10380 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 − 𝐵) = 0) |
13 | 1, 10, 12 | 3eqtr4a 2682 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐵 + 1)...𝐵)) = (𝐵 − 𝐵)) |
14 | oveq1 6657 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 + 1) = (𝐵 + 1)) | |
15 | 14 | oveq1d 6665 | . . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 + 1)...𝐵) = ((𝐵 + 1)...𝐵)) |
16 | 15 | fveq2d 6195 | . . . 4 ⊢ (𝐴 = 𝐵 → (#‘((𝐴 + 1)...𝐵)) = (#‘((𝐵 + 1)...𝐵))) |
17 | oveq2 6658 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 − 𝐴) = (𝐵 − 𝐵)) | |
18 | 16, 17 | eqeq12d 2637 | . . 3 ⊢ (𝐴 = 𝐵 → ((#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴) ↔ (#‘((𝐵 + 1)...𝐵)) = (𝐵 − 𝐵))) |
19 | 13, 18 | syl5ibr 236 | . 2 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴))) |
20 | uzp1 11721 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 = 𝐴 ∨ 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) | |
21 | pm2.24 121 | . . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) | |
22 | 21 | eqcoms 2630 | . . . . . . . 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 394 | . . . . . . 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 446 | . . . . 5 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) |
27 | hashfz 13214 | . . . . 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 11692 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℤ) | |
30 | 29 | zcnd 11483 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℂ) |
31 | 1cnd 10056 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 1 ∈ ℂ) | |
32 | 11, 30, 31 | nppcan2d 10418 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐵 − (𝐴 + 1)) + 1) = (𝐵 − 𝐴)) |
33 | 32 | adantl 482 | . . . 4 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((𝐵 − (𝐴 + 1)) + 1) = (𝐵 − 𝐴)) |
34 | 28, 33 | eqtrd 2656 | . . 3 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴)) |
35 | 34 | ex 450 | . 2 ⊢ (¬ 𝐴 = 𝐵 → (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴))) |
36 | 19, 35 | pm2.61i 176 | 1 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 − cmin 10266 ℤcz 11377 ℤ≥cuz 11687 ...cfz 12326 #chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 |
This theorem is referenced by: 2lgslem1 25119 |
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