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Theorem relexpaddnn 15090
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 7439 . . . . . 6 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
21coeq1d 5872 . . . . 5 (𝑛 = 1 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)))
3 oveq1 7438 . . . . . 6 (𝑛 = 1 → (𝑛 + 𝑀) = (1 + 𝑀))
43oveq2d 7447 . . . . 5 (𝑛 = 1 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(1 + 𝑀)))
52, 4eqeq12d 2753 . . . 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 7439 . . . . . 6 (𝑛 = 𝑘 → (𝑅𝑟𝑛) = (𝑅𝑟𝑘))
87coeq1d 5872 . . . . 5 (𝑛 = 𝑘 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)))
9 oveq1 7438 . . . . . 6 (𝑛 = 𝑘 → (𝑛 + 𝑀) = (𝑘 + 𝑀))
109oveq2d 7447 . . . . 5 (𝑛 = 𝑘 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))
118, 10eqeq12d 2753 . . . 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 7439 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑘 + 1)))
1413coeq1d 5872 . . . . 5 (𝑛 = (𝑘 + 1) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)))
15 oveq1 7438 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑛 + 𝑀) = ((𝑘 + 1) + 𝑀))
1615oveq2d 7447 . . . . 5 (𝑛 = (𝑘 + 1) → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
1714, 16eqeq12d 2753 . . . 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 7439 . . . . . 6 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2019coeq1d 5872 . . . . 5 (𝑛 = 𝑁 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)))
21 oveq1 7438 . . . . . 6 (𝑛 = 𝑁 → (𝑛 + 𝑀) = (𝑁 + 𝑀))
2221oveq2d 7447 . . . . 5 (𝑛 = 𝑁 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
2320, 22eqeq12d 2753 . . . 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 15065 . . . . . 6 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2625adantl 481 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
2726coeq1d 5872 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ (𝑅𝑟𝑀)))
28 relexpsucnnl 15069 . . . . 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 12282 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
32 1cnd 11256 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 1 ∈ ℂ)
3331, 32addcomd 11463 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑀 + 1) = (1 + 𝑀))
3433oveq2d 7447 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑀 + 1)) = (𝑅𝑟(1 + 𝑀)))
3527, 29, 343eqtr2d 2783 . . 3 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(1 + 𝑀)))
36 simp2r 1201 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑅𝑉)
37 simp1 1137 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℕ)
38 relexpsucnnl 15069 . . . . . . . . 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 5872 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = ((𝑅 ∘ (𝑅𝑟𝑘)) ∘ (𝑅𝑟𝑀)))
41 coass 6285 . . . . . . 7 ((𝑅 ∘ (𝑅𝑟𝑘)) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)))
4240, 41eqtrdi 2793 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀))))
43 simp3 1139 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))
4443coeq2d 5873 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀))) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
4537nncnd 12282 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℂ)
46 1cnd 11256 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 1 ∈ ℂ)
47313ad2ant2 1135 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℂ)
4845, 46, 47add32d 11489 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑘 + 1) + 𝑀) = ((𝑘 + 𝑀) + 1))
4948oveq2d 7447 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟((𝑘 + 1) + 𝑀)) = (𝑅𝑟((𝑘 + 𝑀) + 1)))
50303ad2ant2 1135 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℕ)
5137, 50nnaddcld 12318 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑘 + 𝑀) ∈ ℕ)
52 relexpsucnnl 15069 . . . . . . . 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 2778 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
5542, 44, 543eqtrd 2781 . . . . 5 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
56553exp 1120 . . . 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 12284 . 2 (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
59583impib 1117 1 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  ccom 5689  (class class class)co 7431  cc 11153  1c1 11156   + caddc 11158  cn 12266  𝑟crelexp 15058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-relexp 15059
This theorem is referenced by:  relexpaddg  15092  iunrelexpmin1  43721  relexpmulnn  43722  iunrelexpmin2  43725  relexpaddss  43731
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