Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  relexpaddnn Structured version   Visualization version   GIF version

 Description: Relation composition becomes addition under exponentiation. (Contributed by RP, 23-May-2020.)
Assertion
Ref Expression
relexpaddnn ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))

Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6913 . . . . . 6 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
21coeq1d 5516 . . . . 5 (𝑛 = 1 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)))
3 oveq1 6912 . . . . . 6 (𝑛 = 1 → (𝑛 + 𝑀) = (1 + 𝑀))
43oveq2d 6921 . . . . 5 (𝑛 = 1 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(1 + 𝑀)))
52, 4eqeq12d 2840 . . . 4 (𝑛 = 1 → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(1 + 𝑀))))
65imbi2d 332 . . 3 (𝑛 = 1 → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(1 + 𝑀)))))
7 oveq2 6913 . . . . . 6 (𝑛 = 𝑘 → (𝑅𝑟𝑛) = (𝑅𝑟𝑘))
87coeq1d 5516 . . . . 5 (𝑛 = 𝑘 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)))
9 oveq1 6912 . . . . . 6 (𝑛 = 𝑘 → (𝑛 + 𝑀) = (𝑘 + 𝑀))
109oveq2d 6921 . . . . 5 (𝑛 = 𝑘 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))
118, 10eqeq12d 2840 . . . 4 (𝑛 = 𝑘 → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))))
1211imbi2d 332 . . 3 (𝑛 = 𝑘 → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))))
13 oveq2 6913 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑘 + 1)))
1413coeq1d 5516 . . . . 5 (𝑛 = (𝑘 + 1) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)))
15 oveq1 6912 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑛 + 𝑀) = ((𝑘 + 1) + 𝑀))
1615oveq2d 6921 . . . . 5 (𝑛 = (𝑘 + 1) → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
1714, 16eqeq12d 2840 . . . 4 (𝑛 = (𝑘 + 1) → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀))))
1817imbi2d 332 . . 3 (𝑛 = (𝑘 + 1) → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))))
19 oveq2 6913 . . . . . 6 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2019coeq1d 5516 . . . . 5 (𝑛 = 𝑁 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)))
21 oveq1 6912 . . . . . 6 (𝑛 = 𝑁 → (𝑛 + 𝑀) = (𝑁 + 𝑀))
2221oveq2d 6921 . . . . 5 (𝑛 = 𝑁 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
2320, 22eqeq12d 2840 . . . 4 (𝑛 = 𝑁 → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
2423imbi2d 332 . . 3 (𝑛 = 𝑁 → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))))
25 relexp1g 14143 . . . . . 6 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2625adantl 475 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
2726coeq1d 5516 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ (𝑅𝑟𝑀)))
28 relexpsucnnl 14149 . . . . 5 ((𝑅𝑉𝑀 ∈ ℕ) → (𝑅𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅𝑟𝑀)))
2928ancoms 452 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅𝑟𝑀)))
30 simpl 476 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 𝑀 ∈ ℕ)
3130nncnd 11368 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
32 1cnd 10351 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 1 ∈ ℂ)
3331, 32addcomd 10557 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑀 + 1) = (1 + 𝑀))
3433oveq2d 6921 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑀 + 1)) = (𝑅𝑟(1 + 𝑀)))
3527, 29, 343eqtr2d 2867 . . 3 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(1 + 𝑀)))
36 simp2r 1261 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑅𝑉)
37 simp1 1170 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℕ)
38 relexpsucnnl 14149 . . . . . . . . 9 ((𝑅𝑉𝑘 ∈ ℕ) → (𝑅𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅𝑟𝑘)))
3936, 37, 38syl2anc 579 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅𝑟𝑘)))
4039coeq1d 5516 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = ((𝑅 ∘ (𝑅𝑟𝑘)) ∘ (𝑅𝑟𝑀)))
41 coass 5895 . . . . . . 7 ((𝑅 ∘ (𝑅𝑟𝑘)) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)))
4240, 41syl6eq 2877 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀))))
43 simp3 1172 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))
4443coeq2d 5517 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀))) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
4537nncnd 11368 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℂ)
46 1cnd 10351 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 1 ∈ ℂ)
47313ad2ant2 1168 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℂ)
4845, 46, 47add32d 10582 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑘 + 1) + 𝑀) = ((𝑘 + 𝑀) + 1))
4948oveq2d 6921 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟((𝑘 + 1) + 𝑀)) = (𝑅𝑟((𝑘 + 𝑀) + 1)))
50303ad2ant2 1168 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℕ)
5137, 50nnaddcld 11403 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑘 + 𝑀) ∈ ℕ)
52 relexpsucnnl 14149 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑘 + 𝑀) ∈ ℕ) → (𝑅𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
5336, 51, 52syl2anc 579 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
5449, 53eqtr2d 2862 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
5542, 44, 543eqtrd 2865 . . . . 5 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
56553exp 1152 . . . 4 (𝑘 ∈ ℕ → ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀)) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))))
5756a2d 29 . . 3 (𝑘 ∈ ℕ → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))))
586, 12, 18, 24, 35, 57nnind 11370 . 2 (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
59583impib 1148 1 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ∘ ccom 5346  (class class class)co 6905  ℂcc 10250  1c1 10253   + caddc 10255  ℕcn 11350  ↑𝑟crelexp 14137 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-n0 11619  df-z 11705  df-uz 11969  df-seq 13096  df-relexp 14138 This theorem is referenced by:  relexpaddg  14170  iunrelexpmin1  38834  relexpmulnn  38835  iunrelexpmin2  38838  relexpaddss  38844
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