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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fzo0opth | Structured version Visualization version GIF version | ||
| Description: Equality for a half open integer range starting at zero is the same as equality of its upper bound, analogous to fzopth 13601 and fzoopth 13801. (Contributed by Thierry Arnoux, 27-May-2025.) |
| Ref | Expression |
|---|---|
| fzo0opth.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
| fzo0opth.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
| Ref | Expression |
|---|---|
| fzo0opth | ⊢ (𝜑 → ((0..^𝑀) = (0..^𝑁) ↔ 𝑀 = 𝑁)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0z 12624 | . . . 4 ⊢ 0 ∈ ℤ | |
| 2 | fzo0opth.1 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
| 3 | 2 | nn0zd 12639 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 4 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 0 < 𝑀) → 0 < 𝑀) | |
| 5 | fzoopth 13801 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀) → ((0..^𝑀) = (0..^𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁))) | |
| 6 | 1, 3, 4, 5 | mp3an2ani 1470 | . . 3 ⊢ ((𝜑 ∧ 0 < 𝑀) → ((0..^𝑀) = (0..^𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁))) |
| 7 | eqid 2737 | . . . 4 ⊢ 0 = 0 | |
| 8 | 7 | biantrur 530 | . . 3 ⊢ (𝑀 = 𝑁 ↔ (0 = 0 ∧ 𝑀 = 𝑁)) |
| 9 | 6, 8 | bitr4di 289 | . 2 ⊢ ((𝜑 ∧ 0 < 𝑀) → ((0..^𝑀) = (0..^𝑁) ↔ 𝑀 = 𝑁)) |
| 10 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 = 𝑀) → 0 = 𝑀) | |
| 11 | 10 | oveq2d 7447 | . . . . . 6 ⊢ ((𝜑 ∧ 0 = 𝑀) → (0..^0) = (0..^𝑀)) |
| 12 | fzo0 13723 | . . . . . 6 ⊢ (0..^0) = ∅ | |
| 13 | 11, 12 | eqtr3di 2792 | . . . . 5 ⊢ ((𝜑 ∧ 0 = 𝑀) → (0..^𝑀) = ∅) |
| 14 | 13 | eqeq1d 2739 | . . . 4 ⊢ ((𝜑 ∧ 0 = 𝑀) → ((0..^𝑀) = (0..^𝑁) ↔ ∅ = (0..^𝑁))) |
| 15 | eqcom 2744 | . . . 4 ⊢ (∅ = (0..^𝑁) ↔ (0..^𝑁) = ∅) | |
| 16 | 14, 15 | bitrdi 287 | . . 3 ⊢ ((𝜑 ∧ 0 = 𝑀) → ((0..^𝑀) = (0..^𝑁) ↔ (0..^𝑁) = ∅)) |
| 17 | 0zd 12625 | . . . 4 ⊢ ((𝜑 ∧ 0 = 𝑀) → 0 ∈ ℤ) | |
| 18 | fzo0opth.2 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
| 19 | 18 | nn0zd 12639 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 20 | 19 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 0 = 𝑀) → 𝑁 ∈ ℤ) |
| 21 | fzon 13720 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅)) | |
| 22 | 17, 20, 21 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 0 = 𝑀) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅)) |
| 23 | nn0le0eq0 12554 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 0 ↔ 𝑁 = 0)) | |
| 24 | 23 | biimpa 476 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 0) → 𝑁 = 0) |
| 25 | 18, 24 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ≤ 0) → 𝑁 = 0) |
| 26 | 25 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ 0 = 𝑀) ∧ 𝑁 ≤ 0) → 𝑁 = 0) |
| 27 | id 22 | . . . . . . 7 ⊢ (𝑁 = 0 → 𝑁 = 0) | |
| 28 | 0le0 12367 | . . . . . . 7 ⊢ 0 ≤ 0 | |
| 29 | 27, 28 | eqbrtrdi 5182 | . . . . . 6 ⊢ (𝑁 = 0 → 𝑁 ≤ 0) |
| 30 | 29 | adantl 481 | . . . . 5 ⊢ (((𝜑 ∧ 0 = 𝑀) ∧ 𝑁 = 0) → 𝑁 ≤ 0) |
| 31 | 26, 30 | impbida 801 | . . . 4 ⊢ ((𝜑 ∧ 0 = 𝑀) → (𝑁 ≤ 0 ↔ 𝑁 = 0)) |
| 32 | eqcom 2744 | . . . . 5 ⊢ (𝑁 = 0 ↔ 0 = 𝑁) | |
| 33 | 32 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 0 = 𝑀) → (𝑁 = 0 ↔ 0 = 𝑁)) |
| 34 | 10 | eqeq1d 2739 | . . . 4 ⊢ ((𝜑 ∧ 0 = 𝑀) → (0 = 𝑁 ↔ 𝑀 = 𝑁)) |
| 35 | 31, 33, 34 | 3bitrd 305 | . . 3 ⊢ ((𝜑 ∧ 0 = 𝑀) → (𝑁 ≤ 0 ↔ 𝑀 = 𝑁)) |
| 36 | 16, 22, 35 | 3bitr2d 307 | . 2 ⊢ ((𝜑 ∧ 0 = 𝑀) → ((0..^𝑀) = (0..^𝑁) ↔ 𝑀 = 𝑁)) |
| 37 | 2 | nn0ge0d 12590 | . . 3 ⊢ (𝜑 → 0 ≤ 𝑀) |
| 38 | 0red 11264 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 39 | 2 | nn0red 12588 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 40 | 38, 39 | leloed 11404 | . . 3 ⊢ (𝜑 → (0 ≤ 𝑀 ↔ (0 < 𝑀 ∨ 0 = 𝑀))) |
| 41 | 37, 40 | mpbid 232 | . 2 ⊢ (𝜑 → (0 < 𝑀 ∨ 0 = 𝑀)) |
| 42 | 9, 36, 41 | mpjaodan 961 | 1 ⊢ (𝜑 → ((0..^𝑀) = (0..^𝑁) ↔ 𝑀 = 𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∅c0 4333 class class class wbr 5143 (class class class)co 7431 0cc0 11155 < clt 11295 ≤ cle 11296 ℕ0cn0 12526 ℤcz 12613 ..^cfzo 13694 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 |
| This theorem is referenced by: 1arithidomlem2 33564 1arithidom 33565 |
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