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Mirrors > Home > MPE Home > Th. List > elfzonlteqm1 | Structured version Visualization version GIF version |
Description: If an element of a half-open integer range is not less than the upper bound of the range decreased by 1, it must be equal to the upper bound of the range decreased by 1. (Contributed by AV, 3-Nov-2018.) |
Ref | Expression |
---|---|
elfzonlteqm1 | ⊢ ((𝐴 ∈ (0..^𝐵) ∧ ¬ 𝐴 < (𝐵 − 1)) → 𝐴 = (𝐵 − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 12556 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | elfzo0 13660 | . . . . 5 ⊢ (𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) | |
3 | elnnuz 12853 | . . . . . . . 8 ⊢ (𝐵 ∈ ℕ ↔ 𝐵 ∈ (ℤ≥‘1)) | |
4 | 3 | biimpi 215 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ (ℤ≥‘1)) |
5 | 0p1e1 12321 | . . . . . . . . 9 ⊢ (0 + 1) = 1 | |
6 | 5 | a1i 11 | . . . . . . . 8 ⊢ (𝐵 ∈ ℕ → (0 + 1) = 1) |
7 | 6 | fveq2d 6885 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → (ℤ≥‘(0 + 1)) = (ℤ≥‘1)) |
8 | 4, 7 | eleqtrrd 2837 | . . . . . 6 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ (ℤ≥‘(0 + 1))) |
9 | 8 | 3ad2ant2 1135 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵) → 𝐵 ∈ (ℤ≥‘(0 + 1))) |
10 | 2, 9 | sylbi 216 | . . . 4 ⊢ (𝐴 ∈ (0..^𝐵) → 𝐵 ∈ (ℤ≥‘(0 + 1))) |
11 | fzosplitsnm1 13694 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝐵 ∈ (ℤ≥‘(0 + 1))) → (0..^𝐵) = ((0..^(𝐵 − 1)) ∪ {(𝐵 − 1)})) | |
12 | 1, 10, 11 | sylancr 588 | . . 3 ⊢ (𝐴 ∈ (0..^𝐵) → (0..^𝐵) = ((0..^(𝐵 − 1)) ∪ {(𝐵 − 1)})) |
13 | eleq2 2823 | . . . 4 ⊢ ((0..^𝐵) = ((0..^(𝐵 − 1)) ∪ {(𝐵 − 1)}) → (𝐴 ∈ (0..^𝐵) ↔ 𝐴 ∈ ((0..^(𝐵 − 1)) ∪ {(𝐵 − 1)}))) | |
14 | elun 4146 | . . . . 5 ⊢ (𝐴 ∈ ((0..^(𝐵 − 1)) ∪ {(𝐵 − 1)}) ↔ (𝐴 ∈ (0..^(𝐵 − 1)) ∨ 𝐴 ∈ {(𝐵 − 1)})) | |
15 | elfzo0 13660 | . . . . . . 7 ⊢ (𝐴 ∈ (0..^(𝐵 − 1)) ↔ (𝐴 ∈ ℕ0 ∧ (𝐵 − 1) ∈ ℕ ∧ 𝐴 < (𝐵 − 1))) | |
16 | pm2.24 124 | . . . . . . . 8 ⊢ (𝐴 < (𝐵 − 1) → (¬ 𝐴 < (𝐵 − 1) → 𝐴 = (𝐵 − 1))) | |
17 | 16 | 3ad2ant3 1136 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ (𝐵 − 1) ∈ ℕ ∧ 𝐴 < (𝐵 − 1)) → (¬ 𝐴 < (𝐵 − 1) → 𝐴 = (𝐵 − 1))) |
18 | 15, 17 | sylbi 216 | . . . . . 6 ⊢ (𝐴 ∈ (0..^(𝐵 − 1)) → (¬ 𝐴 < (𝐵 − 1) → 𝐴 = (𝐵 − 1))) |
19 | elsni 4641 | . . . . . . 7 ⊢ (𝐴 ∈ {(𝐵 − 1)} → 𝐴 = (𝐵 − 1)) | |
20 | 19 | a1d 25 | . . . . . 6 ⊢ (𝐴 ∈ {(𝐵 − 1)} → (¬ 𝐴 < (𝐵 − 1) → 𝐴 = (𝐵 − 1))) |
21 | 18, 20 | jaoi 856 | . . . . 5 ⊢ ((𝐴 ∈ (0..^(𝐵 − 1)) ∨ 𝐴 ∈ {(𝐵 − 1)}) → (¬ 𝐴 < (𝐵 − 1) → 𝐴 = (𝐵 − 1))) |
22 | 14, 21 | sylbi 216 | . . . 4 ⊢ (𝐴 ∈ ((0..^(𝐵 − 1)) ∪ {(𝐵 − 1)}) → (¬ 𝐴 < (𝐵 − 1) → 𝐴 = (𝐵 − 1))) |
23 | 13, 22 | syl6bi 253 | . . 3 ⊢ ((0..^𝐵) = ((0..^(𝐵 − 1)) ∪ {(𝐵 − 1)}) → (𝐴 ∈ (0..^𝐵) → (¬ 𝐴 < (𝐵 − 1) → 𝐴 = (𝐵 − 1)))) |
24 | 12, 23 | mpcom 38 | . 2 ⊢ (𝐴 ∈ (0..^𝐵) → (¬ 𝐴 < (𝐵 − 1) → 𝐴 = (𝐵 − 1))) |
25 | 24 | imp 408 | 1 ⊢ ((𝐴 ∈ (0..^𝐵) ∧ ¬ 𝐴 < (𝐵 − 1)) → 𝐴 = (𝐵 − 1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∪ cun 3944 {csn 4624 class class class wbr 5144 ‘cfv 6535 (class class class)co 7396 0cc0 11097 1c1 11098 + caddc 11100 < clt 11235 − cmin 11431 ℕcn 12199 ℕ0cn0 12459 ℤcz 12545 ℤ≥cuz 12809 ..^cfzo 13614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 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 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 df-fzo 13615 |
This theorem is referenced by: clwwisshclwwslemlem 29233 |
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