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Theorem relexpaddnn 13911
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 6773 . . . . . 6 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
21coeq1d 5391 . . . . 5 (𝑛 = 1 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)))
3 oveq1 6772 . . . . . 6 (𝑛 = 1 → (𝑛 + 𝑀) = (1 + 𝑀))
43oveq2d 6781 . . . . 5 (𝑛 = 1 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(1 + 𝑀)))
52, 4eqeq12d 2739 . . . 4 (𝑛 = 1 → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(1 + 𝑀))))
65imbi2d 329 . . 3 (𝑛 = 1 → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(1 + 𝑀)))))
7 oveq2 6773 . . . . . 6 (𝑛 = 𝑘 → (𝑅𝑟𝑛) = (𝑅𝑟𝑘))
87coeq1d 5391 . . . . 5 (𝑛 = 𝑘 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)))
9 oveq1 6772 . . . . . 6 (𝑛 = 𝑘 → (𝑛 + 𝑀) = (𝑘 + 𝑀))
109oveq2d 6781 . . . . 5 (𝑛 = 𝑘 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))
118, 10eqeq12d 2739 . . . 4 (𝑛 = 𝑘 → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))))
1211imbi2d 329 . . 3 (𝑛 = 𝑘 → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))))
13 oveq2 6773 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑘 + 1)))
1413coeq1d 5391 . . . . 5 (𝑛 = (𝑘 + 1) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)))
15 oveq1 6772 . . . . . 6 (𝑛 = (𝑘 + 1) → (𝑛 + 𝑀) = ((𝑘 + 1) + 𝑀))
1615oveq2d 6781 . . . . 5 (𝑛 = (𝑘 + 1) → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
1714, 16eqeq12d 2739 . . . 4 (𝑛 = (𝑘 + 1) → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀))))
1817imbi2d 329 . . 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 6773 . . . . . 6 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2019coeq1d 5391 . . . . 5 (𝑛 = 𝑁 → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)))
21 oveq1 6772 . . . . . 6 (𝑛 = 𝑁 → (𝑛 + 𝑀) = (𝑁 + 𝑀))
2221oveq2d 6781 . . . . 5 (𝑛 = 𝑁 → (𝑅𝑟(𝑛 + 𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
2320, 22eqeq12d 2739 . . . 4 (𝑛 = 𝑁 → (((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀)) ↔ ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
2423imbi2d 329 . . 3 (𝑛 = 𝑁 → (((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑛) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))))
25 relexp1g 13886 . . . . . 6 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2625adantl 473 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
2726coeq1d 5391 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ (𝑅𝑟𝑀)))
28 relexpsucnnl 13892 . . . . 5 ((𝑅𝑉𝑀 ∈ ℕ) → (𝑅𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅𝑟𝑀)))
2928ancoms 468 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅𝑟𝑀)))
30 simpl 474 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 𝑀 ∈ ℕ)
3130nncnd 11149 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
32 1cnd 10169 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → 1 ∈ ℂ)
3331, 32addcomd 10351 . . . . 5 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑀 + 1) = (1 + 𝑀))
3433oveq2d 6781 . . . 4 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑀 + 1)) = (𝑅𝑟(1 + 𝑀)))
3527, 29, 343eqtr2d 2764 . . 3 ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟1) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(1 + 𝑀)))
36 simp2r 1219 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑅𝑉)
37 simp1 1128 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℕ)
38 relexpsucnnl 13892 . . . . . . . . 9 ((𝑅𝑉𝑘 ∈ ℕ) → (𝑅𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅𝑟𝑘)))
3936, 37, 38syl2anc 696 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅𝑟𝑘)))
4039coeq1d 5391 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = ((𝑅 ∘ (𝑅𝑟𝑘)) ∘ (𝑅𝑟𝑀)))
41 coass 5767 . . . . . . 7 ((𝑅 ∘ (𝑅𝑟𝑘)) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)))
4240, 41syl6eq 2774 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀))))
43 simp3 1130 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀)))
4443coeq2d 5392 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅 ∘ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀))) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
4537nncnd 11149 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℂ)
46 1cnd 10169 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 1 ∈ ℂ)
47313ad2ant2 1126 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℂ)
4845, 46, 47add32d 10376 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑘 + 1) + 𝑀) = ((𝑘 + 𝑀) + 1))
4948oveq2d 6781 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟((𝑘 + 1) + 𝑀)) = (𝑅𝑟((𝑘 + 𝑀) + 1)))
50303ad2ant2 1126 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℕ)
5137, 50nnaddcld 11180 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑘 + 𝑀) ∈ ℕ)
52 relexpsucnnl 13892 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑘 + 𝑀) ∈ ℕ) → (𝑅𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
5336, 51, 52syl2anc 696 . . . . . . 7 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))))
5449, 53eqtr2d 2759 . . . . . 6 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → (𝑅 ∘ (𝑅𝑟(𝑘 + 𝑀))) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
5542, 44, 543eqtrd 2762 . . . . 5 ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅𝑉) ∧ ((𝑅𝑟𝑘) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑘 + 𝑀))) → ((𝑅𝑟(𝑘 + 1)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟((𝑘 + 1) + 𝑀)))
56553exp 1112 . . . 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 11151 . 2 (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))
59583impib 1108 1 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1596  wcel 2103  ccom 5222  (class class class)co 6765  cc 10047  1c1 10050   + caddc 10052  cn 11133  𝑟crelexp 13880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-n0 11406  df-z 11491  df-uz 11801  df-seq 12917  df-relexp 13881
This theorem is referenced by:  relexpaddg  13913  iunrelexpmin1  38419  relexpmulnn  38420  iunrelexpmin2  38423  relexpaddss  38429
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