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

Theorem relexpsucnnl 14977
Description: A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
Assertion
Ref Expression
relexpsucnnl ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))

Proof of Theorem relexpsucnnl
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7416 . . . . . 6 (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
21oveq2d 7425 . . . . 5 (𝑛 = 1 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(1 + 1)))
3 oveq2 7417 . . . . . 6 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
43coeq2d 5863 . . . . 5 (𝑛 = 1 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟1)))
52, 4eqeq12d 2749 . . . 4 (𝑛 = 1 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1))))
65imbi2d 341 . . 3 (𝑛 = 1 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1)))))
7 oveq1 7416 . . . . . 6 (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
87oveq2d 7425 . . . . 5 (𝑛 = 𝑚 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(𝑚 + 1)))
9 oveq2 7417 . . . . . 6 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
109coeq2d 5863 . . . . 5 (𝑛 = 𝑚 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟𝑚)))
118, 10eqeq12d 2749 . . . 4 (𝑛 = 𝑚 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))))
1211imbi2d 341 . . 3 (𝑛 = 𝑚 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))))
13 oveq1 7416 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑛 + 1) = ((𝑚 + 1) + 1))
1413oveq2d 7425 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟((𝑚 + 1) + 1)))
15 oveq2 7417 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1615coeq2d 5863 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))
1714, 16eqeq12d 2749 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1)))))
1817imbi2d 341 . . 3 (𝑛 = (𝑚 + 1) → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))))
19 oveq1 7416 . . . . . 6 (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
2019oveq2d 7425 . . . . 5 (𝑛 = 𝑁 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(𝑁 + 1)))
21 oveq2 7417 . . . . . 6 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2221coeq2d 5863 . . . . 5 (𝑛 = 𝑁 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟𝑁)))
2320, 22eqeq12d 2749 . . . 4 (𝑛 = 𝑁 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁))))
2423imbi2d 341 . . 3 (𝑛 = 𝑁 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))))
25 relexp1g 14973 . . . . 5 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2625coeq1d 5862 . . . 4 (𝑅𝑉 → ((𝑅𝑟1) ∘ 𝑅) = (𝑅𝑅))
27 1nn 12223 . . . . 5 1 ∈ ℕ
28 relexpsucnnr 14972 . . . . 5 ((𝑅𝑉 ∧ 1 ∈ ℕ) → (𝑅𝑟(1 + 1)) = ((𝑅𝑟1) ∘ 𝑅))
2927, 28mpan2 690 . . . 4 (𝑅𝑉 → (𝑅𝑟(1 + 1)) = ((𝑅𝑟1) ∘ 𝑅))
3025coeq2d 5863 . . . 4 (𝑅𝑉 → (𝑅 ∘ (𝑅𝑟1)) = (𝑅𝑅))
3126, 29, 303eqtr4d 2783 . . 3 (𝑅𝑉 → (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1)))
32 coeq1 5858 . . . . . . . . 9 ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = ((𝑅 ∘ (𝑅𝑟𝑚)) ∘ 𝑅))
33 coass 6265 . . . . . . . . 9 ((𝑅 ∘ (𝑅𝑟𝑚)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅))
3432, 33eqtrdi 2789 . . . . . . . 8 ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
3534adantl 483 . . . . . . 7 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
36 simpl 484 . . . . . . . 8 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑉𝑚 ∈ ℕ))
37 peano2nn 12224 . . . . . . . . 9 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
3837anim2i 618 . . . . . . . 8 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑉 ∧ (𝑚 + 1) ∈ ℕ))
39 relexpsucnnr 14972 . . . . . . . 8 ((𝑅𝑉 ∧ (𝑚 + 1) ∈ ℕ) → (𝑅𝑟((𝑚 + 1) + 1)) = ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅))
4036, 38, 393syl 18 . . . . . . 7 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑟((𝑚 + 1) + 1)) = ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅))
41 relexpsucnnr 14972 . . . . . . . . 9 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
4241adantr 482 . . . . . . . 8 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
4342coeq2d 5863 . . . . . . 7 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅 ∘ (𝑅𝑟(𝑚 + 1))) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
4435, 40, 433eqtr4d 2783 . . . . . 6 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))
4544ex 414 . . . . 5 ((𝑅𝑉𝑚 ∈ ℕ) → ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1)))))
4645expcom 415 . . . 4 (𝑚 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))))
4746a2d 29 . . 3 (𝑚 ∈ ℕ → ((𝑅𝑉 → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑉 → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))))
486, 12, 18, 24, 31, 47nnind 12230 . 2 (𝑁 ∈ ℕ → (𝑅𝑉 → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁))))
4948impcom 409 1 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  ccom 5681  (class class class)co 7409  1c1 11111   + caddc 11113  cn 12212  𝑟crelexp 14966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-seq 13967  df-relexp 14967
This theorem is referenced by:  relexpsucl  14978  relexpcnv  14982  relexpaddnn  14998  trclfvcom  42474  trclimalb2  42477
  Copyright terms: Public domain W3C validator