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Theorem relexprelg 14393
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 11891 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 eqeq1 2805 . . . . . . . 8 (𝑛 = 1 → (𝑛 = 1 ↔ 1 = 1))
32imbi1d 345 . . . . . . 7 (𝑛 = 1 → ((𝑛 = 1 → Rel 𝑅) ↔ (1 = 1 → Rel 𝑅)))
43anbi2d 631 . . . . . 6 (𝑛 = 1 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (1 = 1 → Rel 𝑅))))
5 oveq2 7147 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
65releqd 5621 . . . . . 6 (𝑛 = 1 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟1)))
74, 6imbi12d 348 . . . . 5 (𝑛 = 1 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅𝑟1))))
8 eqeq1 2805 . . . . . . . 8 (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
98imbi1d 345 . . . . . . 7 (𝑛 = 𝑚 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑚 = 1 → Rel 𝑅)))
109anbi2d 631 . . . . . 6 (𝑛 = 𝑚 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (𝑚 = 1 → Rel 𝑅))))
11 oveq2 7147 . . . . . . 7 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
1211releqd 5621 . . . . . 6 (𝑛 = 𝑚 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟𝑚)))
1310, 12imbi12d 348 . . . . 5 (𝑛 = 𝑚 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (𝑚 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑚))))
14 eqeq1 2805 . . . . . . . 8 (𝑛 = (𝑚 + 1) → (𝑛 = 1 ↔ (𝑚 + 1) = 1))
1514imbi1d 345 . . . . . . 7 (𝑛 = (𝑚 + 1) → ((𝑛 = 1 → Rel 𝑅) ↔ ((𝑚 + 1) = 1 → Rel 𝑅)))
1615anbi2d 631 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅))))
17 oveq2 7147 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1817releqd 5621 . . . . . 6 (𝑛 = (𝑚 + 1) → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟(𝑚 + 1))))
1916, 18imbi12d 348 . . . . 5 (𝑛 = (𝑚 + 1) → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅)) → Rel (𝑅𝑟(𝑚 + 1)))))
20 eqeq1 2805 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 = 1 ↔ 𝑁 = 1))
2120imbi1d 345 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑁 = 1 → Rel 𝑅)))
2221anbi2d 631 . . . . . 6 (𝑛 = 𝑁 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅))))
23 oveq2 7147 . . . . . . 7 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2423releqd 5621 . . . . . 6 (𝑛 = 𝑁 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟𝑁)))
2522, 24imbi12d 348 . . . . 5 (𝑛 = 𝑁 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))))
26 eqid 2801 . . . . . . . 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 485 . . . . . 6 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel 𝑅)
30 relexp1g 14381 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
3130adantr 484 . . . . . . 7 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → (𝑅𝑟1) = 𝑅)
3231releqd 5621 . . . . . 6 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → (Rel (𝑅𝑟1) ↔ Rel 𝑅))
3329, 32mpbird 260 . . . . 5 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅𝑟1))
34 relco 6068 . . . . . . . . 9 Rel ((𝑅𝑟𝑚) ∘ 𝑅)
35 relexpsucnnr 14380 . . . . . . . . . . 11 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3635ancoms 462 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3736releqd 5621 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (Rel (𝑅𝑟(𝑚 + 1)) ↔ Rel ((𝑅𝑟𝑚) ∘ 𝑅)))
3834, 37mpbiri 261 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → Rel (𝑅𝑟(𝑚 + 1)))
3938a1d 25 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (((𝑚 + 1) = 1 → Rel 𝑅) → Rel (𝑅𝑟(𝑚 + 1))))
4039expimpd 457 . . . . . 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 11647 . . . 4 (𝑁 ∈ ℕ → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
43 relexp0rel 14392 . . . . . . . 8 (𝑅𝑉 → Rel (𝑅𝑟0))
4443adantl 485 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟0))
45 simpl 486 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
4645oveq2d 7155 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
4746releqd 5621 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (Rel (𝑅𝑟𝑁) ↔ Rel (𝑅𝑟0)))
4844, 47mpbird 260 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
4948a1d 25 . . . . 5 ((𝑁 = 0 ∧ 𝑅𝑉) → ((𝑁 = 1 → Rel 𝑅) → Rel (𝑅𝑟𝑁)))
5049expimpd 457 . . . 4 (𝑁 = 0 → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
5142, 50jaoi 854 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
521, 51sylbi 220 . 2 (𝑁 ∈ ℕ0 → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
53523impib 1113 1 ((𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  ccom 5527  Rel wrel 5528  (class class class)co 7139  0cc0 10530  1c1 10531   + caddc 10533  cn 11629  0cn0 11889  𝑟crelexp 14374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-seq 13369  df-relexp 14375
This theorem is referenced by:  relexprel  14394  relexpfld  14404  relexpuzrel  14407
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