Mathbox for metakunt |
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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 6706 | . . . . . . 7 ⊢ (𝑋 = 𝐼 → (𝐴‘𝑋) = (𝐴‘𝐼)) | |
4 | 3 | adantl 485 | . . . . . 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 488 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → 𝑥 = 𝐼) | |
8 | 7 | iftrued 4437 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐼) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑀) |
9 | 1zzd 12191 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ) | |
10 | metakunt5.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
11 | 10 | nnzd 12264 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
12 | metakunt5.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ ℕ) | |
13 | 12 | nnzd 12264 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
14 | 12 | nnge1d 11861 | . . . . . . . . 9 ⊢ (𝜑 → 1 ≤ 𝐼) |
15 | metakunt5.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
16 | 9, 11, 13, 14, 15 | elfzd 13086 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (1...𝑀)) |
17 | 6, 8, 16, 10 | fvmptd 6814 | . . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐼) = 𝑀) |
18 | 17 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐴‘𝐼) = 𝑀) |
19 | 4, 18 | eqtrd 2774 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐴‘𝑋) = 𝑀) |
20 | 19 | eqeq2d 2745 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝑦 = (𝐴‘𝑋) ↔ 𝑦 = 𝑀)) |
21 | iftrue 4435 | . . . . . . 7 ⊢ (𝑦 = 𝑀 → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝐼) | |
22 | 21 | 3ad2ant3 1137 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝐼 ∧ 𝑦 = 𝑀) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝐼) |
23 | simp2 1139 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝐼 ∧ 𝑦 = 𝑀) → 𝑋 = 𝐼) | |
24 | 22, 23 | eqtr4d 2777 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝐼 ∧ 𝑦 = 𝑀) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝑋) |
25 | 24 | 3expia 1123 | . . . 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 410 | . 2 ⊢ (((𝜑 ∧ 𝑋 = 𝐼) ∧ 𝑦 = (𝐴‘𝑋)) → if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))) = 𝑋) |
28 | 10, 12, 15, 5 | metakunt1 39799 | . . . 4 ⊢ (𝜑 → 𝐴:(1...𝑀)⟶(1...𝑀)) |
29 | 28 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝐴:(1...𝑀)⟶(1...𝑀)) |
30 | metakunt5.6 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) | |
31 | 30 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → 𝑋 ∈ (1...𝑀)) |
32 | 29, 31 | ffvelrnd 6894 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐴‘𝑋) ∈ (1...𝑀)) |
33 | 2, 27, 32, 31 | fvmptd 6814 | 1 ⊢ ((𝜑 ∧ 𝑋 = 𝐼) → (𝐶‘(𝐴‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ifcif 4429 class class class wbr 5043 ↦ cmpt 5124 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 1c1 10713 + caddc 10715 < clt 10850 ≤ cle 10851 − cmin 11045 ℕcn 11813 ...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-uz 12422 df-fz 13079 |
This theorem is referenced by: metakunt9 39807 |
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