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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 14352 | . . . 4 ⊢ (♯‘∅) = 0 | |
2 | eluzelre 12857 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℝ) | |
3 | 2 | ltp1d 12168 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 < (𝐵 + 1)) |
4 | eluzelz 12856 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
5 | peano2z 12627 | . . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (𝐵 + 1) ∈ ℤ) | |
6 | 5 | ancri 549 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → ((𝐵 + 1) ∈ ℤ ∧ 𝐵 ∈ ℤ)) |
7 | fzn 13543 | . . . . . . 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 231 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐵 + 1)...𝐵) = ∅) |
10 | 9 | fveq2d 6895 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘((𝐵 + 1)...𝐵)) = (♯‘∅)) |
11 | 4 | zcnd 12691 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℂ) |
12 | 11 | subidd 11583 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 − 𝐵) = 0) |
13 | 1, 10, 12 | 3eqtr4a 2794 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘((𝐵 + 1)...𝐵)) = (𝐵 − 𝐵)) |
14 | oveq1 7421 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 + 1) = (𝐵 + 1)) | |
15 | 14 | fvoveq1d 7436 | . . . 4 ⊢ (𝐴 = 𝐵 → (♯‘((𝐴 + 1)...𝐵)) = (♯‘((𝐵 + 1)...𝐵))) |
16 | oveq2 7422 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 − 𝐴) = (𝐵 − 𝐵)) | |
17 | 15, 16 | eqeq12d 2744 | . . 3 ⊢ (𝐴 = 𝐵 → ((♯‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴) ↔ (♯‘((𝐵 + 1)...𝐵)) = (𝐵 − 𝐵))) |
18 | 13, 17 | imbitrrid 245 | . 2 ⊢ (𝐴 = 𝐵 → (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴))) |
19 | uzp1 12887 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐵 = 𝐴 ∨ 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) | |
20 | pm2.24 124 | . . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) | |
21 | 20 | eqcoms 2736 | . . . . . . . 8 ⊢ (𝐵 = 𝐴 → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) |
22 | ax-1 6 | . . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘(𝐴 + 1)) → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) | |
23 | 21, 22 | jaoi 856 | . . . . . . 7 ⊢ ((𝐵 = 𝐴 ∨ 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) |
24 | 19, 23 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (¬ 𝐴 = 𝐵 → 𝐵 ∈ (ℤ≥‘(𝐴 + 1)))) |
25 | 24 | impcom 407 | . . . . 5 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐵 ∈ (ℤ≥‘(𝐴 + 1))) |
26 | hashfz 14412 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘(𝐴 + 1)) → (♯‘((𝐴 + 1)...𝐵)) = ((𝐵 − (𝐴 + 1)) + 1)) | |
27 | 25, 26 | syl 17 | . . . 4 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (♯‘((𝐴 + 1)...𝐵)) = ((𝐵 − (𝐴 + 1)) + 1)) |
28 | eluzel2 12851 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℤ) | |
29 | 28 | zcnd 12691 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℂ) |
30 | 1cnd 11233 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 1 ∈ ℂ) | |
31 | 11, 29, 30 | nppcan2d 11621 | . . . . 5 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → ((𝐵 − (𝐴 + 1)) + 1) = (𝐵 − 𝐴)) |
32 | 31 | adantl 481 | . . . 4 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((𝐵 − (𝐴 + 1)) + 1) = (𝐵 − 𝐴)) |
33 | 27, 32 | eqtrd 2768 | . . 3 ⊢ ((¬ 𝐴 = 𝐵 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (♯‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴)) |
34 | 33 | ex 412 | . 2 ⊢ (¬ 𝐴 = 𝐵 → (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴))) |
35 | 18, 34 | pm2.61i 182 | 1 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ∅c0 4318 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 0cc0 11132 1c1 11133 + caddc 11135 < clt 11272 − cmin 11468 ℤcz 12582 ℤ≥cuz 12846 ...cfz 13510 ♯chash 14315 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-hash 14316 |
This theorem is referenced by: 2lgslem1 27320 sticksstones12a 41623 |
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