Mathbox for metakunt |
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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 2745 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 = 𝑋 ↔ 𝑥 = 𝐼)) |
5 | simpr 488 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 = 𝐼) | |
6 | 5 | iftrued 4437 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀) |
7 | 6 | ex 416 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 = 𝐼 → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀)) |
8 | 4, 7 | sylbid 243 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 = 𝑋 → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀)) |
9 | 8 | imp 410 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀) |
10 | 1zzd 12191 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | metakunt26.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
12 | nnz 12182 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
13 | 11, 12 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
14 | metakunt26.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ) | |
15 | 14 | nnzd 12264 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
16 | 14 | nnge1d 11861 | . . . . . . . . 9 ⊢ (𝜑 → 1 ≤ 𝐼) |
17 | metakunt26.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
18 | 10, 13, 15, 16, 17 | elfzd 13086 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
19 | 3 | eleq1d 2818 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (1...𝑀) ↔ 𝐼 ∈ (1...𝑀))) |
20 | 18, 19 | mpbird 260 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) |
21 | 2, 9, 20, 11 | fvmptd 6814 | . . . . . 6 ⊢ (𝜑 → (𝐴‘𝑋) = 𝑀) |
22 | 21 | fveq2d 6710 | . . . . 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 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑀) → 𝑧 = 𝑀) | |
26 | 25 | iftrued 4437 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑀) → if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))) = 𝑀) |
27 | 1zzd 12191 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 1 ∈ ℤ) | |
28 | nnge1 11841 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀) | |
29 | nnre 11820 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
30 | 29 | leidd 11381 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ≤ 𝑀) |
31 | 27, 12, 12, 28, 30 | elfzd 13086 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (1...𝑀)) |
32 | 11, 31 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀)) |
33 | 24, 26, 32, 11 | fvmptd 6814 | . . . . 5 ⊢ (𝜑 → (𝐵‘𝑀) = 𝑀) |
34 | 22, 33 | eqtrd 2774 | . . . 4 ⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = 𝑀) |
35 | 34 | fveq2d 6710 | . . 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 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → 𝑦 = 𝑀) | |
39 | 38 | iftrued 4437 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝐼) |
40 | 37, 39, 32, 14 | fvmptd 6814 | . . 3 ⊢ (𝜑 → (𝐶‘𝑀) = 𝐼) |
41 | 35, 40 | eqtrd 2774 | . 2 ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝐼) |
42 | 3 | eqcomd 2740 | . 2 ⊢ (𝜑 → 𝐼 = 𝑋) |
43 | 41, 42 | eqtrd 2774 | 1 ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ifcif 4429 class class class wbr 5043 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 1c1 10713 + caddc 10715 < clt 10850 ≤ cle 10851 − cmin 11045 ℕcn 11813 ℤcz 12159 ...cfz 13078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-fz 13079 |
This theorem is referenced by: metakunt31 39829 |
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