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Theorem relexprelg 15074
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 12526 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 eqeq1 2739 . . . . . . . 8 (𝑛 = 1 → (𝑛 = 1 ↔ 1 = 1))
32imbi1d 341 . . . . . . 7 (𝑛 = 1 → ((𝑛 = 1 → Rel 𝑅) ↔ (1 = 1 → Rel 𝑅)))
43anbi2d 630 . . . . . 6 (𝑛 = 1 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (1 = 1 → Rel 𝑅))))
5 oveq2 7439 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
65releqd 5791 . . . . . 6 (𝑛 = 1 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟1)))
74, 6imbi12d 344 . . . . 5 (𝑛 = 1 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅𝑟1))))
8 eqeq1 2739 . . . . . . . 8 (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
98imbi1d 341 . . . . . . 7 (𝑛 = 𝑚 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑚 = 1 → Rel 𝑅)))
109anbi2d 630 . . . . . 6 (𝑛 = 𝑚 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (𝑚 = 1 → Rel 𝑅))))
11 oveq2 7439 . . . . . . 7 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
1211releqd 5791 . . . . . 6 (𝑛 = 𝑚 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟𝑚)))
1310, 12imbi12d 344 . . . . 5 (𝑛 = 𝑚 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (𝑚 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑚))))
14 eqeq1 2739 . . . . . . . 8 (𝑛 = (𝑚 + 1) → (𝑛 = 1 ↔ (𝑚 + 1) = 1))
1514imbi1d 341 . . . . . . 7 (𝑛 = (𝑚 + 1) → ((𝑛 = 1 → Rel 𝑅) ↔ ((𝑚 + 1) = 1 → Rel 𝑅)))
1615anbi2d 630 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅))))
17 oveq2 7439 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1817releqd 5791 . . . . . 6 (𝑛 = (𝑚 + 1) → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟(𝑚 + 1))))
1916, 18imbi12d 344 . . . . 5 (𝑛 = (𝑚 + 1) → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅)) → Rel (𝑅𝑟(𝑚 + 1)))))
20 eqeq1 2739 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 = 1 ↔ 𝑁 = 1))
2120imbi1d 341 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑁 = 1 → Rel 𝑅)))
2221anbi2d 630 . . . . . 6 (𝑛 = 𝑁 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅))))
23 oveq2 7439 . . . . . . 7 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2423releqd 5791 . . . . . 6 (𝑛 = 𝑁 → (Rel (𝑅𝑟𝑛) ↔ Rel (𝑅𝑟𝑁)))
2522, 24imbi12d 344 . . . . 5 (𝑛 = 𝑁 → (((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑛)) ↔ ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))))
26 eqid 2735 . . . . . . . 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 481 . . . . . 6 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel 𝑅)
30 relexp1g 15062 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
3130adantr 480 . . . . . . 7 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → (𝑅𝑟1) = 𝑅)
3231releqd 5791 . . . . . 6 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → (Rel (𝑅𝑟1) ↔ Rel 𝑅))
3329, 32mpbird 257 . . . . 5 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅𝑟1))
34 relco 6129 . . . . . . . . 9 Rel ((𝑅𝑟𝑚) ∘ 𝑅)
35 relexpsucnnr 15061 . . . . . . . . . . 11 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3635ancoms 458 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3736releqd 5791 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (Rel (𝑅𝑟(𝑚 + 1)) ↔ Rel ((𝑅𝑟𝑚) ∘ 𝑅)))
3834, 37mpbiri 258 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → Rel (𝑅𝑟(𝑚 + 1)))
3938a1d 25 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (((𝑚 + 1) = 1 → Rel 𝑅) → Rel (𝑅𝑟(𝑚 + 1))))
4039expimpd 453 . . . . . 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 12282 . . . 4 (𝑁 ∈ ℕ → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
43 relexp0rel 15073 . . . . . . . 8 (𝑅𝑉 → Rel (𝑅𝑟0))
4443adantl 481 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟0))
45 simpl 482 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
4645oveq2d 7447 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
4746releqd 5791 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (Rel (𝑅𝑟𝑁) ↔ Rel (𝑅𝑟0)))
4844, 47mpbird 257 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
4948a1d 25 . . . . 5 ((𝑁 = 0 ∧ 𝑅𝑉) → ((𝑁 = 1 → Rel 𝑅) → Rel (𝑅𝑟𝑁)))
5049expimpd 453 . . . 4 (𝑁 = 0 → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
5142, 50jaoi 857 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
521, 51sylbi 217 . 2 (𝑁 ∈ ℕ0 → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
53523impib 1115 1 ((𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  ccom 5693  Rel wrel 5694  (class class class)co 7431  0cc0 11153  1c1 11154   + caddc 11156  cn 12264  0cn0 12524  𝑟crelexp 15055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040  df-relexp 15056
This theorem is referenced by:  relexprel  15075  relexpfld  15085  relexpuzrel  15088
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