MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relexprelg Structured version   Visualization version   GIF version

Theorem relexprelg 15018
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 12505 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 eqeq1 2732 . . . . . . . 8 (𝑛 = 1 → (𝑛 = 1 ↔ 1 = 1))
32imbi1d 341 . . . . . . 7 (𝑛 = 1 → ((𝑛 = 1 → Rel 𝑅) ↔ (1 = 1 → Rel 𝑅)))
43anbi2d 629 . . . . . 6 (𝑛 = 1 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (1 = 1 → Rel 𝑅))))
5 oveq2 7428 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
65releqd 5780 . . . . . 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 2732 . . . . . . . 8 (𝑛 = 𝑚 → (𝑛 = 1 ↔ 𝑚 = 1))
98imbi1d 341 . . . . . . 7 (𝑛 = 𝑚 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑚 = 1 → Rel 𝑅)))
109anbi2d 629 . . . . . 6 (𝑛 = 𝑚 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (𝑚 = 1 → Rel 𝑅))))
11 oveq2 7428 . . . . . . 7 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
1211releqd 5780 . . . . . 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 2732 . . . . . . . 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 629 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ ((𝑚 + 1) = 1 → Rel 𝑅))))
17 oveq2 7428 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1817releqd 5780 . . . . . 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 2732 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛 = 1 ↔ 𝑁 = 1))
2120imbi1d 341 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛 = 1 → Rel 𝑅) ↔ (𝑁 = 1 → Rel 𝑅)))
2221anbi2d 629 . . . . . 6 (𝑛 = 𝑁 → ((𝑅𝑉 ∧ (𝑛 = 1 → Rel 𝑅)) ↔ (𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅))))
23 oveq2 7428 . . . . . . 7 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2423releqd 5780 . . . . . 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 2728 . . . . . . . 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 15006 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
3130adantr 480 . . . . . . 7 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → (𝑅𝑟1) = 𝑅)
3231releqd 5780 . . . . . 6 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → (Rel (𝑅𝑟1) ↔ Rel 𝑅))
3329, 32mpbird 257 . . . . 5 ((𝑅𝑉 ∧ (1 = 1 → Rel 𝑅)) → Rel (𝑅𝑟1))
34 relco 6112 . . . . . . . . 9 Rel ((𝑅𝑟𝑚) ∘ 𝑅)
35 relexpsucnnr 15005 . . . . . . . . . . 11 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3635ancoms 458 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3736releqd 5780 . . . . . . . . 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 12261 . . . 4 (𝑁 ∈ ℕ → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
43 relexp0rel 15017 . . . . . . . 8 (𝑅𝑉 → Rel (𝑅𝑟0))
4443adantl 481 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟0))
45 simpl 482 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
4645oveq2d 7436 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
4746releqd 5780 . . . . . . 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 856 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
521, 51sylbi 216 . 2 (𝑁 ∈ ℕ0 → ((𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁)))
53523impib 1114 1 ((𝑁 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  ccom 5682  Rel wrel 5683  (class class class)co 7420  0cc0 11139  1c1 11140   + caddc 11142  cn 12243  0cn0 12503  𝑟crelexp 14999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-n0 12504  df-z 12590  df-uz 12854  df-seq 14000  df-relexp 15000
This theorem is referenced by:  relexprel  15019  relexpfld  15029  relexpuzrel  15032
  Copyright terms: Public domain W3C validator