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Theorem relexprelg 14261
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 11712 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 eqeq1 2782 . . . . . . . 8 (𝑛 = 1 → (𝑛 = 1 ↔ 1 = 1))
32imbi1d 334 . . . . . . 7 (𝑛 = 1 → ((𝑛 = 1 → Rel 𝑅) ↔ (1 = 1 → Rel 𝑅)))
43anbi2d 619 . . . . . 6 (𝑛 = 1 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (1 = 1 → Rel 𝑅))))
5 oveq2 6986 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
65releqd 5504 . . . . . 6 (𝑛 = 1 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟1)))
74, 6imbi12d 337 . . . . 5 (𝑛 = 1 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅𝑟1))))
8 eqeq1 2782 . . . . . . . 8 (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
98imbi1d 334 . . . . . . 7 (𝑛 = 𝑚 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑚 = 1 → Rel 𝑅)))
109anbi2d 619 . . . . . 6 (𝑛 = 𝑚 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (𝑚 = 1 → Rel 𝑅))))
11 oveq2 6986 . . . . . . 7 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
1211releqd 5504 . . . . . 6 (𝑛 = 𝑚 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟𝑚)))
1310, 12imbi12d 337 . . . . 5 (𝑛 = 𝑚 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (𝑚 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑚))))
14 eqeq1 2782 . . . . . . . 8 (𝑛 = (𝑚 + 1) → (𝑛 = 1 ↔ (𝑚 + 1) = 1))
1514imbi1d 334 . . . . . . 7 (𝑛 = (𝑚 + 1) → ((𝑛 = 1 → Rel 𝑅) ↔ ((𝑚 + 1) = 1 → Rel 𝑅)))
1615anbi2d 619 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅))))
17 oveq2 6986 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1817releqd 5504 . . . . . 6 (𝑛 = (𝑚 + 1) → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟(𝑚 + 1))))
1916, 18imbi12d 337 . . . . 5 (𝑛 = (𝑚 + 1) → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅)) → Rel (𝑅𝑟(𝑚 + 1)))))
20 eqeq1 2782 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 = 1 ↔ 𝑁 = 1))
2120imbi1d 334 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑁 = 1 → Rel 𝑅)))
2221anbi2d 619 . . . . . 6 (𝑛 = 𝑁 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅))))
23 oveq2 6986 . . . . . . 7 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2423releqd 5504 . . . . . 6 (𝑛 = 𝑁 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟𝑁)))
2522, 24imbi12d 337 . . . . 5 (𝑛 = 𝑁 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))))
26 eqid 2778 . . . . . . . 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 474 . . . . . 6 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel 𝑅)
30 relexp1g 14249 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
3130adantr 473 . . . . . . 7 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → (𝑅𝑟1) = 𝑅)
3231releqd 5504 . . . . . 6 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → (Rel (𝑅𝑟1) ↔ Rel 𝑅))
3329, 32mpbird 249 . . . . 5 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅𝑟1))
34 relco 5938 . . . . . . . . 9 Rel ((𝑅𝑟𝑚) ∘ 𝑅)
35 relexpsucnnr 14248 . . . . . . . . . . 11 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3635ancoms 451 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3736releqd 5504 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (Rel (𝑅𝑟(𝑚 + 1)) ↔ Rel ((𝑅𝑟𝑚) ∘ 𝑅)))
3834, 37mpbiri 250 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → Rel (𝑅𝑟(𝑚 + 1)))
3938a1d 25 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (((𝑚 + 1) = 1 → Rel 𝑅) → Rel (𝑅𝑟(𝑚 + 1))))
4039expimpd 446 . . . . . 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 11461 . . . 4 (𝑁 ∈ ℕ → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
43 relexp0rel 14260 . . . . . . . 8 (𝑅𝑉 → Rel (𝑅𝑟0))
4443adantl 474 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟0))
45 simpl 475 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
4645oveq2d 6994 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
4746releqd 5504 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (Rel (𝑅𝑟𝑁) ↔ Rel (𝑅𝑟0)))
4844, 47mpbird 249 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
4948a1d 25 . . . . 5 ((𝑁 = 0 ∧ 𝑅𝑉) → ((𝑁 = 1 → Rel 𝑅) → Rel (𝑅𝑟𝑁)))
5049expimpd 446 . . . 4 (𝑁 = 0 → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
5142, 50jaoi 843 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
521, 51sylbi 209 . 2 (𝑁 ∈ ℕ0 → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
53523impib 1096 1 ((𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wo 833  w3a 1068   = wceq 1507  wcel 2050  ccom 5412  Rel wrel 5413  (class class class)co 6978  0cc0 10337  1c1 10338   + caddc 10340  cn 11441  0cn0 11710  𝑟crelexp 14243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-icn 10396  ax-addcl 10397  ax-addrcl 10398  ax-mulcl 10399  ax-mulrcl 10400  ax-mulcom 10401  ax-addass 10402  ax-mulass 10403  ax-distr 10404  ax-i2m1 10405  ax-1ne0 10406  ax-1rid 10407  ax-rnegex 10408  ax-rrecex 10409  ax-cnre 10410  ax-pre-lttri 10411  ax-pre-lttrn 10412  ax-pre-ltadd 10413  ax-pre-mulgt0 10414
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-er 8091  df-en 8309  df-dom 8310  df-sdom 8311  df-pnf 10478  df-mnf 10479  df-xr 10480  df-ltxr 10481  df-le 10482  df-sub 10674  df-neg 10675  df-nn 11442  df-n0 11711  df-z 11797  df-uz 12062  df-seq 13188  df-relexp 14244
This theorem is referenced by:  relexprel  14262  relexpfld  14272  relexpuzrel  14275
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