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Mirrors > Home > MPE Home > Th. List > Mathboxes > metakunt26 | Structured version Visualization version GIF version |
Description: Construction of one solution of the increment equation system. (Contributed by metakunt, 7-Jul-2024.) |
Ref | Expression |
---|---|
metakunt26.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
metakunt26.2 | ⊢ (𝜑 → 𝐼 ∈ ℕ) |
metakunt26.3 | ⊢ (𝜑 → 𝐼 ≤ 𝑀) |
metakunt26.4 | ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) |
metakunt26.5 | ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) |
metakunt26.6 | ⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) |
metakunt26.7 | ⊢ (𝜑 → 𝑋 = 𝐼) |
Ref | Expression |
---|---|
metakunt26 | ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metakunt26.4 | . . . . . . . 8 ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))) |
3 | metakunt26.7 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = 𝐼) | |
4 | 3 | eqeq2d 2747 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 = 𝑋 ↔ 𝑥 = 𝐼)) |
5 | simpr 485 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 = 𝐼) | |
6 | 5 | iftrued 4494 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀) |
7 | 6 | ex 413 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 = 𝐼 → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀)) |
8 | 4, 7 | sylbid 239 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 = 𝑋 → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀)) |
9 | 8 | imp 407 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀) |
10 | 1zzd 12534 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | metakunt26.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
12 | nnz 12520 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
13 | 11, 12 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
14 | metakunt26.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ) | |
15 | 14 | nnzd 12526 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
16 | 14 | nnge1d 12201 | . . . . . . . . 9 ⊢ (𝜑 → 1 ≤ 𝐼) |
17 | metakunt26.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
18 | 10, 13, 15, 16, 17 | elfzd 13432 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
19 | 3 | eleq1d 2822 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (1...𝑀) ↔ 𝐼 ∈ (1...𝑀))) |
20 | 18, 19 | mpbird 256 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) |
21 | 2, 9, 20, 11 | fvmptd 6955 | . . . . . 6 ⊢ (𝜑 → (𝐴‘𝑋) = 𝑀) |
22 | 21 | fveq2d 6846 | . . . . 5 ⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = (𝐵‘𝑀)) |
23 | metakunt26.6 | . . . . . . 7 ⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))))) |
25 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑀) → 𝑧 = 𝑀) | |
26 | 25 | iftrued 4494 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑀) → if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))) = 𝑀) |
27 | 1zzd 12534 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 1 ∈ ℤ) | |
28 | nnge1 12181 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀) | |
29 | nnre 12160 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
30 | 29 | leidd 11721 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ≤ 𝑀) |
31 | 27, 12, 12, 28, 30 | elfzd 13432 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (1...𝑀)) |
32 | 11, 31 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀)) |
33 | 24, 26, 32, 11 | fvmptd 6955 | . . . . 5 ⊢ (𝜑 → (𝐵‘𝑀) = 𝑀) |
34 | 22, 33 | eqtrd 2776 | . . . 4 ⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = 𝑀) |
35 | 34 | fveq2d 6846 | . . 3 ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = (𝐶‘𝑀)) |
36 | metakunt26.5 | . . . . 5 ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) | |
37 | 36 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))) |
38 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → 𝑦 = 𝑀) | |
39 | 38 | iftrued 4494 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝐼) |
40 | 37, 39, 32, 14 | fvmptd 6955 | . . 3 ⊢ (𝜑 → (𝐶‘𝑀) = 𝐼) |
41 | 35, 40 | eqtrd 2776 | . 2 ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝐼) |
42 | 3 | eqcomd 2742 | . 2 ⊢ (𝜑 → 𝐼 = 𝑋) |
43 | 41, 42 | eqtrd 2776 | 1 ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ifcif 4486 class class class wbr 5105 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 − cmin 11385 ℕcn 12153 ℤcz 12499 ...cfz 13424 |
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 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
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 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-fz 13425 |
This theorem is referenced by: metakunt31 40607 |
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