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Theorem relexpaddnn 15086
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 7438 . . . . . 6 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
21coeq1d 5874 . . . . 5 (𝑛 = 1 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)))
3 oveq1 7437 . . . . . 6 (𝑛 = 1 → (𝑛 + 𝑀) = (1 + 𝑀))
43oveq2d 7446 . . . . 5 (𝑛 = 1 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(1 + 𝑀)))
52, 4eqeq12d 2750 . . . 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 7438 . . . . . 6 (𝑛 = 𝑘 → (𝑅𝑟𝑛) = (𝑅𝑟𝑘))
87coeq1d 5874 . . . . 5 (𝑛 = 𝑘 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)))
9 oveq1 7437 . . . . . 6 (𝑛 = 𝑘 → (𝑛 + 𝑀) = (𝑘 + 𝑀))
109oveq2d 7446 . . . . 5 (𝑛 = 𝑘 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))
118, 10eqeq12d 2750 . . . 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 7438 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑘 + 1)))
1413coeq1d 5874 . . . . 5 (𝑛 = (𝑘 + 1) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)))
15 oveq1 7437 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑛 + 𝑀) = ((𝑘 + 1) + 𝑀))
1615oveq2d 7446 . . . . 5 (𝑛 = (𝑘 + 1) → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
1714, 16eqeq12d 2750 . . . 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 7438 . . . . . 6 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2019coeq1d 5874 . . . . 5 (𝑛 = 𝑁 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)))
21 oveq1 7437 . . . . . 6 (𝑛 = 𝑁 → (𝑛 + 𝑀) = (𝑁 + 𝑀))
2221oveq2d 7446 . . . . 5 (𝑛 = 𝑁 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
2320, 22eqeq12d 2750 . . . 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 15061 . . . . . 6 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2625adantl 481 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
2726coeq1d 5874 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ (𝑅𝑟𝑀)))
28 relexpsucnnl 15065 . . . . 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 12279 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
32 1cnd 11253 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 1 ∈ ℂ)
3331, 32addcomd 11460 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑀 + 1) = (1 + 𝑀))
3433oveq2d 7446 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑀 + 1)) = (𝑅𝑟(1 + 𝑀)))
3527, 29, 343eqtr2d 2780 . . 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 15065 . . . . . . . . 9 ((𝑅𝑉𝑘 ∈ ℕ) → (𝑅𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅𝑟𝑘)))
3936, 37, 38syl2anc 584 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅𝑟𝑘)))
4039coeq1d 5874 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = ((𝑅 ∘ (𝑅𝑟𝑘)) ∘ (𝑅𝑟𝑀)))
41 coass 6286 . . . . . . 7 ((𝑅 ∘ (𝑅𝑟𝑘)) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)))
4240, 41eqtrdi 2790 . . . . . 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 5875 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀))) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
4537nncnd 12279 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℂ)
46 1cnd 11253 . . . . . . . . 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 11486 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑘 + 1) + 𝑀) = ((𝑘 + 𝑀) + 1))
4948oveq2d 7446 . . . . . . 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 12315 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑘 + 𝑀) ∈ ℕ)
52 relexpsucnnl 15065 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑘 + 𝑀) ∈ ℕ) → (𝑅𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
5336, 51, 52syl2anc 584 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
5449, 53eqtr2d 2775 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
5542, 44, 543eqtrd 2778 . . . . 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 12281 . 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 1536  wcel 2105  ccom 5692  (class class class)co 7430  cc 11150  1c1 11153   + caddc 11155  cn 12263  𝑟crelexp 15054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-seq 14039  df-relexp 15055
This theorem is referenced by:  relexpaddg  15088  iunrelexpmin1  43697  relexpmulnn  43698  iunrelexpmin2  43701  relexpaddss  43707
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