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Mirrors > Home > MPE Home > Th. List > Mathboxes > fzom1ne1 | Structured version Visualization version GIF version |
Description: Elementhood in a half-open interval, except the lower bound, shifted by one. (Contributed by Thierry Arnoux, 1-Jan-2024.) |
Ref | Expression |
---|---|
fzom1ne1 | ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → (𝐾 − 1) ∈ (𝑀..^(𝑁 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzone1 30659 | . . 3 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → 𝐾 ∈ ((𝑀 + 1)..^𝑁)) | |
2 | 1z 12065 | . . 3 ⊢ 1 ∈ ℤ | |
3 | fzosubel 13159 | . . 3 ⊢ ((𝐾 ∈ ((𝑀 + 1)..^𝑁) ∧ 1 ∈ ℤ) → (𝐾 − 1) ∈ (((𝑀 + 1) − 1)..^(𝑁 − 1))) | |
4 | 1, 2, 3 | sylancl 589 | . 2 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → (𝐾 − 1) ∈ (((𝑀 + 1) − 1)..^(𝑁 − 1))) |
5 | elfzoel1 13099 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ) | |
6 | 5 | adantr 484 | . . . . 5 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → 𝑀 ∈ ℤ) |
7 | 6 | zcnd 12141 | . . . 4 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → 𝑀 ∈ ℂ) |
8 | 1cnd 10688 | . . . 4 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → 1 ∈ ℂ) | |
9 | 7, 8 | pncand 11050 | . . 3 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → ((𝑀 + 1) − 1) = 𝑀) |
10 | 9 | oveq1d 7172 | . 2 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → (((𝑀 + 1) − 1)..^(𝑁 − 1)) = (𝑀..^(𝑁 − 1))) |
11 | 4, 10 | eleqtrd 2855 | 1 ⊢ ((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → (𝐾 − 1) ∈ (𝑀..^(𝑁 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2112 ≠ wne 2952 (class class class)co 7157 1c1 10590 + caddc 10592 − cmin 10922 ℤcz 12034 ..^cfzo 13096 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5174 ax-nul 5181 ax-pow 5239 ax-pr 5303 ax-un 7466 ax-cnex 10645 ax-resscn 10646 ax-1cn 10647 ax-icn 10648 ax-addcl 10649 ax-addrcl 10650 ax-mulcl 10651 ax-mulrcl 10652 ax-mulcom 10653 ax-addass 10654 ax-mulass 10655 ax-distr 10656 ax-i2m1 10657 ax-1ne0 10658 ax-1rid 10659 ax-rnegex 10660 ax-rrecex 10661 ax-cnre 10662 ax-pre-lttri 10663 ax-pre-lttrn 10664 ax-pre-ltadd 10665 ax-pre-mulgt0 10666 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rab 3080 df-v 3412 df-sbc 3700 df-csb 3809 df-dif 3864 df-un 3866 df-in 3868 df-ss 3878 df-pss 3880 df-nul 4229 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4803 df-iun 4889 df-br 5038 df-opab 5100 df-mpt 5118 df-tr 5144 df-id 5435 df-eprel 5440 df-po 5448 df-so 5449 df-fr 5488 df-we 5490 df-xp 5535 df-rel 5536 df-cnv 5537 df-co 5538 df-dm 5539 df-rn 5540 df-res 5541 df-ima 5542 df-pred 6132 df-ord 6178 df-on 6179 df-lim 6180 df-suc 6181 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7115 df-ov 7160 df-oprab 7161 df-mpo 7162 df-om 7587 df-1st 7700 df-2nd 7701 df-wrecs 7964 df-recs 8025 df-rdg 8063 df-er 8306 df-en 8542 df-dom 8543 df-sdom 8544 df-pnf 10729 df-mnf 10730 df-xr 10731 df-ltxr 10732 df-le 10733 df-sub 10924 df-neg 10925 df-nn 11689 df-n0 11949 df-z 12035 df-uz 12297 df-fz 12954 df-fzo 13097 |
This theorem is referenced by: cycpmco2lem6 30938 |
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