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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 2744 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 = 𝑋 ↔ 𝑥 = 𝐼)) |
5 | simpr 486 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 = 𝐼) | |
6 | 5 | iftrued 4495 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀) |
7 | 6 | ex 414 | . . . . . . . . 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 408 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀) |
10 | 1zzd 12539 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | metakunt26.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
12 | nnz 12525 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
13 | 11, 12 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
14 | metakunt26.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ) | |
15 | 14 | nnzd 12531 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
16 | 14 | nnge1d 12206 | . . . . . . . . 9 ⊢ (𝜑 → 1 ≤ 𝐼) |
17 | metakunt26.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
18 | 10, 13, 15, 16, 17 | elfzd 13438 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
19 | 3 | eleq1d 2819 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (1...𝑀) ↔ 𝐼 ∈ (1...𝑀))) |
20 | 18, 19 | mpbird 257 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) |
21 | 2, 9, 20, 11 | fvmptd 6956 | . . . . . 6 ⊢ (𝜑 → (𝐴‘𝑋) = 𝑀) |
22 | 21 | fveq2d 6847 | . . . . 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 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑀) → 𝑧 = 𝑀) | |
26 | 25 | iftrued 4495 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑀) → if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))) = 𝑀) |
27 | 1zzd 12539 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 1 ∈ ℤ) | |
28 | nnge1 12186 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀) | |
29 | nnre 12165 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
30 | 29 | leidd 11726 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ≤ 𝑀) |
31 | 27, 12, 12, 28, 30 | elfzd 13438 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (1...𝑀)) |
32 | 11, 31 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀)) |
33 | 24, 26, 32, 11 | fvmptd 6956 | . . . . 5 ⊢ (𝜑 → (𝐵‘𝑀) = 𝑀) |
34 | 22, 33 | eqtrd 2773 | . . . 4 ⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = 𝑀) |
35 | 34 | fveq2d 6847 | . . 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 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → 𝑦 = 𝑀) | |
39 | 38 | iftrued 4495 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝐼) |
40 | 37, 39, 32, 14 | fvmptd 6956 | . . 3 ⊢ (𝜑 → (𝐶‘𝑀) = 𝐼) |
41 | 35, 40 | eqtrd 2773 | . 2 ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝐼) |
42 | 3 | eqcomd 2739 | . 2 ⊢ (𝜑 → 𝐼 = 𝑋) |
43 | 41, 42 | eqtrd 2773 | 1 ⊢ (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ifcif 4487 class class class wbr 5106 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 1c1 11057 + caddc 11059 < clt 11194 ≤ cle 11195 − cmin 11390 ℕcn 12158 ℤcz 12504 ...cfz 13430 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-fz 13431 |
This theorem is referenced by: metakunt31 40653 |
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