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Mathbox for metakunt |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > metakunt3 | Structured version Visualization version GIF version |
Description: Value of A. (Contributed by metakunt, 23-May-2024.) |
Ref | Expression |
---|---|
metakunt3.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
metakunt3.2 | ⊢ (𝜑 → 𝐼 ∈ ℕ) |
metakunt3.3 | ⊢ (𝜑 → 𝐼 ≤ 𝑀) |
metakunt3.4 | ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) |
metakunt3.5 | ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) |
Ref | Expression |
---|---|
metakunt3 | ⊢ (𝜑 → (𝐴‘𝑋) = if(𝑋 = 𝐼, 𝑀, if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metakunt3.4 | . . 3 ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))) |
3 | eqeq1 2740 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝐼 ↔ 𝑋 = 𝐼)) | |
4 | breq1 5106 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 < 𝐼 ↔ 𝑋 < 𝐼)) | |
5 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
6 | oveq1 7358 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 − 1) = (𝑋 − 1)) | |
7 | 4, 5, 6 | ifbieq12d 4512 | . . . 4 ⊢ (𝑥 = 𝑋 → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = if(𝑋 < 𝐼, 𝑋, (𝑋 − 1))) |
8 | 3, 7 | ifbieq2d 4510 | . . 3 ⊢ (𝑥 = 𝑋 → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if(𝑋 = 𝐼, 𝑀, if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)))) |
9 | 8 | adantl 482 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if(𝑋 = 𝐼, 𝑀, if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)))) |
10 | metakunt3.5 | . 2 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) | |
11 | metakunt3.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
12 | 11 | nnzd 12484 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
13 | 10 | elfzelzd 13396 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℤ) |
14 | 1zzd 12492 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
15 | 13, 14 | zsubcld 12570 | . . . 4 ⊢ (𝜑 → (𝑋 − 1) ∈ ℤ) |
16 | 13, 15 | ifcld 4530 | . . 3 ⊢ (𝜑 → if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)) ∈ ℤ) |
17 | 12, 16 | ifcld 4530 | . 2 ⊢ (𝜑 → if(𝑋 = 𝐼, 𝑀, if(𝑋 < 𝐼, 𝑋, (𝑋 − 1))) ∈ ℤ) |
18 | 2, 9, 10, 17 | fvmptd 6952 | 1 ⊢ (𝜑 → (𝐴‘𝑋) = if(𝑋 = 𝐼, 𝑀, if(𝑋 < 𝐼, 𝑋, (𝑋 − 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ifcif 4484 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6493 (class class class)co 7351 1c1 11010 < clt 11147 ≤ cle 11148 − cmin 11343 ℕcn 12111 ℤcz 12457 ...cfz 13378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 |
This theorem is referenced by: (None) |
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