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Mirrors > Home > MPE Home > Th. List > Mathboxes > metakunt14 | Structured version Visualization version GIF version |
Description: A is a primitive permutation that moves the I-th element to the end and C is its inverse that moves the last element back to the I-th position. (Contributed by metakunt, 25-May-2024.) |
Ref | Expression |
---|---|
metakunt14.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
metakunt14.2 | ⊢ (𝜑 → 𝐼 ∈ ℕ) |
metakunt14.3 | ⊢ (𝜑 → 𝐼 ≤ 𝑀) |
metakunt14.4 | ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) |
metakunt14.5 | ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) |
Ref | Expression |
---|---|
metakunt14 | ⊢ (𝜑 → (𝐴:(1...𝑀)–1-1-onto→(1...𝑀) ∧ ◡𝐴 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metakunt14.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | metakunt14.2 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ℕ) | |
3 | metakunt14.3 | . . . 4 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
4 | metakunt14.4 | . . . 4 ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) | |
5 | 1, 2, 3, 4 | metakunt1 40623 | . . 3 ⊢ (𝜑 → 𝐴:(1...𝑀)⟶(1...𝑀)) |
6 | metakunt14.5 | . . . 4 ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) | |
7 | 1, 2, 3, 6 | metakunt2 40624 | . . 3 ⊢ (𝜑 → 𝐶:(1...𝑀)⟶(1...𝑀)) |
8 | 1 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝑀)) → 𝑀 ∈ ℕ) |
9 | 2 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝑀)) → 𝐼 ∈ ℕ) |
10 | 3 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝑀)) → 𝐼 ≤ 𝑀) |
11 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝑀)) → 𝑎 ∈ (1...𝑀)) | |
12 | 8, 9, 10, 4, 6, 11 | metakunt9 40631 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (1...𝑀)) → (𝐶‘(𝐴‘𝑎)) = 𝑎) |
13 | 12 | ralrimiva 3140 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝑀)(𝐶‘(𝐴‘𝑎)) = 𝑎) |
14 | 1 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑀)) → 𝑀 ∈ ℕ) |
15 | 2 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑀)) → 𝐼 ∈ ℕ) |
16 | 3 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑀)) → 𝐼 ≤ 𝑀) |
17 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑀)) → 𝑏 ∈ (1...𝑀)) | |
18 | 14, 15, 16, 4, 6, 17 | metakunt13 40635 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑀)) → (𝐴‘(𝐶‘𝑏)) = 𝑏) |
19 | 18 | ralrimiva 3140 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ (1...𝑀)(𝐴‘(𝐶‘𝑏)) = 𝑏) |
20 | 5, 7, 13, 19 | 2fvidf1od 7245 | . 2 ⊢ (𝜑 → 𝐴:(1...𝑀)–1-1-onto→(1...𝑀)) |
21 | 5, 7, 13, 19 | 2fvidinvd 7246 | . 2 ⊢ (𝜑 → ◡𝐴 = 𝐶) |
22 | 20, 21 | jca 513 | 1 ⊢ (𝜑 → (𝐴:(1...𝑀)–1-1-onto→(1...𝑀) ∧ ◡𝐴 = 𝐶)) |
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 ◡ccnv 5633 –1-1-onto→wf1o 6496 ‘cfv 6497 (class class class)co 7358 1c1 11057 + caddc 11059 < clt 11194 ≤ cle 11195 − cmin 11390 ℕcn 12158 ...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-uz 12769 df-fz 13431 |
This theorem is referenced by: metakunt34 40656 |
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