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Theorem relexprelg 13985
 Description: The exponentiation of a class is a relation except when the exponent is one and the class is not a relation. (Contributed by RP, 23-May-2020.)
Assertion
Ref Expression
relexprelg ((𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))

Proof of Theorem relexprelg
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn0 11495 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 eqeq1 2774 . . . . . . . 8 (𝑛 = 1 → (𝑛 = 1 ↔ 1 = 1))
32imbi1d 330 . . . . . . 7 (𝑛 = 1 → ((𝑛 = 1 → Rel 𝑅) ↔ (1 = 1 → Rel 𝑅)))
43anbi2d 606 . . . . . 6 (𝑛 = 1 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (1 = 1 → Rel 𝑅))))
5 oveq2 6800 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
65releqd 5343 . . . . . 6 (𝑛 = 1 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟1)))
74, 6imbi12d 333 . . . . 5 (𝑛 = 1 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅𝑟1))))
8 eqeq1 2774 . . . . . . . 8 (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
98imbi1d 330 . . . . . . 7 (𝑛 = 𝑚 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑚 = 1 → Rel 𝑅)))
109anbi2d 606 . . . . . 6 (𝑛 = 𝑚 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (𝑚 = 1 → Rel 𝑅))))
11 oveq2 6800 . . . . . . 7 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
1211releqd 5343 . . . . . 6 (𝑛 = 𝑚 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟𝑚)))
1310, 12imbi12d 333 . . . . 5 (𝑛 = 𝑚 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (𝑚 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑚))))
14 eqeq1 2774 . . . . . . . 8 (𝑛 = (𝑚 + 1) → (𝑛 = 1 ↔ (𝑚 + 1) = 1))
1514imbi1d 330 . . . . . . 7 (𝑛 = (𝑚 + 1) → ((𝑛 = 1 → Rel 𝑅) ↔ ((𝑚 + 1) = 1 → Rel 𝑅)))
1615anbi2d 606 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅))))
17 oveq2 6800 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1817releqd 5343 . . . . . 6 (𝑛 = (𝑚 + 1) → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟(𝑚 + 1))))
1916, 18imbi12d 333 . . . . 5 (𝑛 = (𝑚 + 1) → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅)) → Rel (𝑅𝑟(𝑚 + 1)))))
20 eqeq1 2774 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 = 1 ↔ 𝑁 = 1))
2120imbi1d 330 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑁 = 1 → Rel 𝑅)))
2221anbi2d 606 . . . . . 6 (𝑛 = 𝑁 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅))))
23 oveq2 6800 . . . . . . 7 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2423releqd 5343 . . . . . 6 (𝑛 = 𝑁 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟𝑁)))
2522, 24imbi12d 333 . . . . 5 (𝑛 = 𝑁 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))))
26 eqid 2770 . . . . . . . 8 1 = 1
27 pm2.27 42 . . . . . . . 8 (1 = 1 → ((1 = 1 → Rel 𝑅) → Rel 𝑅))
2826, 27ax-mp 5 . . . . . . 7 ((1 = 1 → Rel 𝑅) → Rel 𝑅)
2928adantl 467 . . . . . 6 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel 𝑅)
30 relexp1g 13973 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
3130adantr 466 . . . . . . 7 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → (𝑅𝑟1) = 𝑅)
3231releqd 5343 . . . . . 6 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → (Rel (𝑅𝑟1) ↔ Rel 𝑅))
3329, 32mpbird 247 . . . . 5 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅𝑟1))
34 relco 5777 . . . . . . . . 9 Rel ((𝑅𝑟𝑚) ∘ 𝑅)
35 relexpsucnnr 13972 . . . . . . . . . . 11 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3635ancoms 455 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3736releqd 5343 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (Rel (𝑅𝑟(𝑚 + 1)) ↔ Rel ((𝑅𝑟𝑚) ∘ 𝑅)))
3834, 37mpbiri 248 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → Rel (𝑅𝑟(𝑚 + 1)))
3938a1d 25 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (((𝑚 + 1) = 1 → Rel 𝑅) → Rel (𝑅𝑟(𝑚 + 1))))
4039expimpd 441 . . . . . 6 (𝑚 ∈ ℕ → ((𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅)) → Rel (𝑅𝑟(𝑚 + 1))))
4140a1d 25 . . . . 5 (𝑚 ∈ ℕ → (((𝑅𝑉 ∧ (𝑚 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑚)) → ((𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅)) → Rel (𝑅𝑟(𝑚 + 1)))))
427, 13, 19, 25, 33, 41nnind 11239 . . . 4 (𝑁 ∈ ℕ → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
43 relexp0rel 13984 . . . . . . . 8 (𝑅𝑉 → Rel (𝑅𝑟0))
4443adantl 467 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟0))
45 simpl 468 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
4645oveq2d 6808 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
4746releqd 5343 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (Rel (𝑅𝑟𝑁) ↔ Rel (𝑅𝑟0)))
4844, 47mpbird 247 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
4948a1d 25 . . . . 5 ((𝑁 = 0 ∧ 𝑅𝑉) → ((𝑁 = 1 → Rel 𝑅) → Rel (𝑅𝑟𝑁)))
5049expimpd 441 . . . 4 (𝑁 = 0 → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
5142, 50jaoi 837 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
521, 51sylbi 207 . 2 (𝑁 ∈ ℕ0 → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
53523impib 1107 1 ((𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∨ wo 826   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144   ∘ ccom 5253  Rel wrel 5254  (class class class)co 6792  0cc0 10137  1c1 10138   + caddc 10140  ℕcn 11221  ℕ0cn0 11493  ↑𝑟crelexp 13967 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-n0 11494  df-z 11579  df-uz 11888  df-seq 13008  df-relexp 13968 This theorem is referenced by:  relexprel  13986  relexpfld  13996  relexpuzrel  13999
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