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Mirrors > Home > MPE Home > Th. List > Mathboxes > metakunt5 | Structured version Visualization version GIF version |
Description: C is the left inverse for A. (Contributed by metakunt, 24-May-2024.) |
Ref | Expression |
---|---|
metakunt5.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
metakunt5.2 | ⊢ (𝜑 → 𝐼 ∈ ℕ) |
metakunt5.3 | ⊢ (𝜑 → 𝐼 ≤ 𝑀) |
metakunt5.4 | ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) |
metakunt5.5 | ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) |
metakunt5.6 | ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) |
Ref | Expression |
---|---|
metakunt5 | ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐶‘(𝐴‘𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metakunt5.5 | . . 3 ⊢ 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1)))) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))) |
3 | fveq2 6651 | . . . . . . 7 ⊢ (𝑋 = 𝐼 → (𝐴‘𝑋) = (𝐴‘𝐼)) | |
4 | 3 | adantl 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐴‘𝑋) = (𝐴‘𝐼)) |
5 | metakunt5.4 | . . . . . . . . 9 ⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) | |
6 | 5 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))) |
7 | simpr 489 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 = 𝐼) | |
8 | 7 | iftrued 4421 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀) |
9 | 1zzd 12037 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ) | |
10 | metakunt5.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
11 | 10 | nnzd 12110 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
12 | metakunt5.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ) | |
13 | 12 | nnzd 12110 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
14 | 12 | nnge1d 11707 | . . . . . . . . 9 ⊢ (𝜑 → 1 ≤ 𝐼) |
15 | metakunt5.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
16 | 9, 11, 13, 14, 15 | elfzd 12932 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
17 | 6, 8, 16, 10 | fvmptd 6759 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐼) = 𝑀) |
18 | 17 | adantr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐴‘𝐼) = 𝑀) |
19 | 4, 18 | eqtrd 2794 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐴‘𝑋) = 𝑀) |
20 | 19 | eqeq2d 2770 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝑦 = (𝐴‘𝑋) ↔ 𝑦 = 𝑀)) |
21 | iftrue 4419 | . . . . . . 7 ⊢ (𝑦 = 𝑀 → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝐼) | |
22 | 21 | 3ad2ant3 1133 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝐼 ∧ 𝑦 = 𝑀) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝐼) |
23 | simp2 1135 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝐼 ∧ 𝑦 = 𝑀) → 𝑋 = 𝐼) | |
24 | 22, 23 | eqtr4d 2797 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼 ∧ 𝑦 = 𝑀) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝑋) |
25 | 24 | 3expia 1119 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝑦 = 𝑀 → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝑋)) |
26 | 20, 25 | sylbid 243 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝑦 = (𝐴‘𝑋) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝑋)) |
27 | 26 | imp 411 | . 2 ⊢ (((𝜑 ∧ 𝑋 = 𝐼) ∧ 𝑦 = (𝐴‘𝑋)) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝑋) |
28 | 10, 12, 15, 5 | metakunt1 39632 | . . . 4 ⊢ (𝜑 → 𝐴:(1...𝑀)⟶(1...𝑀)) |
29 | 28 | adantr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝐴:(1...𝑀)⟶(1...𝑀)) |
30 | metakunt5.6 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) | |
31 | 30 | adantr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝑋 ∈ (1...𝑀)) |
32 | 29, 31 | ffvelrnd 6836 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐴‘𝑋) ∈ (1...𝑀)) |
33 | 2, 27, 32, 31 | fvmptd 6759 | 1 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐶‘(𝐴‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ifcif 4413 class class class wbr 5025 ↦ cmpt 5105 ⟶wf 6324 ‘cfv 6328 (class class class)co 7143 1c1 10561 + caddc 10563 < clt 10698 ≤ cle 10699 − cmin 10893 ℕcn 11659 ...cfz 12924 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-n0 11920 df-z 12006 df-uz 12268 df-fz 12925 |
This theorem is referenced by: metakunt9 39640 |
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