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Theorem relexpaddnn 15003
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)))

Proof of Theorem relexpaddnn
Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7420 . . . . . 6 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
21coeq1d 5861 . . . . 5 (𝑛 = 1 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)))
3 oveq1 7419 . . . . . 6 (𝑛 = 1 → (𝑛 + 𝑀) = (1 + 𝑀))
43oveq2d 7428 . . . . 5 (𝑛 = 1 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(1 + 𝑀)))
52, 4eqeq12d 2747 . . . 4 (𝑛 = 1 → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(1 + 𝑀))))
65imbi2d 340 . . 3 (𝑛 = 1 → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(1 + 𝑀)))))
7 oveq2 7420 . . . . . 6 (𝑛 = 𝑘 → (𝑅𝑟𝑛) = (𝑅𝑟𝑘))
87coeq1d 5861 . . . . 5 (𝑛 = 𝑘 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)))
9 oveq1 7419 . . . . . 6 (𝑛 = 𝑘 → (𝑛 + 𝑀) = (𝑘 + 𝑀))
109oveq2d 7428 . . . . 5 (𝑛 = 𝑘 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))
118, 10eqeq12d 2747 . . . 4 (𝑛 = 𝑘 → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))))
1211imbi2d 340 . . 3 (𝑛 = 𝑘 → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))))
13 oveq2 7420 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑘 + 1)))
1413coeq1d 5861 . . . . 5 (𝑛 = (𝑘 + 1) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)))
15 oveq1 7419 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑛 + 𝑀) = ((𝑘 + 1) + 𝑀))
1615oveq2d 7428 . . . . 5 (𝑛 = (𝑘 + 1) → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
1714, 16eqeq12d 2747 . . . 4 (𝑛 = (𝑘 + 1) → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀))))
1817imbi2d 340 . . 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 7420 . . . . . 6 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2019coeq1d 5861 . . . . 5 (𝑛 = 𝑁 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)))
21 oveq1 7419 . . . . . 6 (𝑛 = 𝑁 → (𝑛 + 𝑀) = (𝑁 + 𝑀))
2221oveq2d 7428 . . . . 5 (𝑛 = 𝑁 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
2320, 22eqeq12d 2747 . . . 4 (𝑛 = 𝑁 → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
2423imbi2d 340 . . 3 (𝑛 = 𝑁 → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))))
25 relexp1g 14978 . . . . . 6 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2625adantl 481 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
2726coeq1d 5861 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ (𝑅𝑟𝑀)))
28 relexpsucnnl 14982 . . . . 5 ((𝑅𝑉𝑀 ∈ ℕ) → (𝑅𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅𝑟𝑀)))
2928ancoms 458 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅𝑟𝑀)))
30 simpl 482 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 𝑀 ∈ ℕ)
3130nncnd 12233 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
32 1cnd 11214 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 1 ∈ ℂ)
3331, 32addcomd 11421 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑀 + 1) = (1 + 𝑀))
3433oveq2d 7428 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑀 + 1)) = (𝑅𝑟(1 + 𝑀)))
3527, 29, 343eqtr2d 2777 . . 3 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(1 + 𝑀)))
36 simp2r 1199 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑅𝑉)
37 simp1 1135 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℕ)
38 relexpsucnnl 14982 . . . . . . . . 9 ((𝑅𝑉𝑘 ∈ ℕ) → (𝑅𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅𝑟𝑘)))
3936, 37, 38syl2anc 583 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅𝑟𝑘)))
4039coeq1d 5861 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = ((𝑅 ∘ (𝑅𝑟𝑘)) ∘ (𝑅𝑟𝑀)))
41 coass 6264 . . . . . . 7 ((𝑅 ∘ (𝑅𝑟𝑘)) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)))
4240, 41eqtrdi 2787 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀))))
43 simp3 1137 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))
4443coeq2d 5862 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀))) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
4537nncnd 12233 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℂ)
46 1cnd 11214 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 1 ∈ ℂ)
47313ad2ant2 1133 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℂ)
4845, 46, 47add32d 11446 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑘 + 1) + 𝑀) = ((𝑘 + 𝑀) + 1))
4948oveq2d 7428 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟((𝑘 + 1) + 𝑀)) = (𝑅𝑟((𝑘 + 𝑀) + 1)))
50303ad2ant2 1133 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℕ)
5137, 50nnaddcld 12269 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑘 + 𝑀) ∈ ℕ)
52 relexpsucnnl 14982 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑘 + 𝑀) ∈ ℕ) → (𝑅𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
5336, 51, 52syl2anc 583 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
5449, 53eqtr2d 2772 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
5542, 44, 543eqtrd 2775 . . . . 5 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
56553exp 1118 . . . 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 12235 . 2 (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
59583impib 1115 1 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  ccom 5680  (class class class)co 7412  cc 11112  1c1 11115   + caddc 11117  cn 12217  𝑟crelexp 14971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-uz 12828  df-seq 13972  df-relexp 14972
This theorem is referenced by:  relexpaddg  15005  iunrelexpmin1  42762  relexpmulnn  42763  iunrelexpmin2  42766  relexpaddss  42772
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