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

Theorem relexpsucnnl 15066
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 7438 . . . . . 6 (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
21oveq2d 7447 . . . . 5 (𝑛 = 1 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(1 + 1)))
3 oveq2 7439 . . . . . 6 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
43coeq2d 5876 . . . . 5 (𝑛 = 1 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟1)))
52, 4eqeq12d 2751 . . . 4 (𝑛 = 1 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1))))
65imbi2d 340 . . 3 (𝑛 = 1 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1)))))
7 oveq1 7438 . . . . . 6 (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
87oveq2d 7447 . . . . 5 (𝑛 = 𝑚 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(𝑚 + 1)))
9 oveq2 7439 . . . . . 6 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
109coeq2d 5876 . . . . 5 (𝑛 = 𝑚 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟𝑚)))
118, 10eqeq12d 2751 . . . 4 (𝑛 = 𝑚 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))))
1211imbi2d 340 . . 3 (𝑛 = 𝑚 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))))
13 oveq1 7438 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑛 + 1) = ((𝑚 + 1) + 1))
1413oveq2d 7447 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟((𝑚 + 1) + 1)))
15 oveq2 7439 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1615coeq2d 5876 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))
1714, 16eqeq12d 2751 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1)))))
1817imbi2d 340 . . 3 (𝑛 = (𝑚 + 1) → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))))
19 oveq1 7438 . . . . . 6 (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
2019oveq2d 7447 . . . . 5 (𝑛 = 𝑁 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(𝑁 + 1)))
21 oveq2 7439 . . . . . 6 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2221coeq2d 5876 . . . . 5 (𝑛 = 𝑁 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟𝑁)))
2320, 22eqeq12d 2751 . . . 4 (𝑛 = 𝑁 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁))))
2423imbi2d 340 . . 3 (𝑛 = 𝑁 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))))
25 relexp1g 15062 . . . . 5 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2625coeq1d 5875 . . . 4 (𝑅𝑉 → ((𝑅𝑟1) ∘ 𝑅) = (𝑅𝑅))
27 1nn 12275 . . . . 5 1 ∈ ℕ
28 relexpsucnnr 15061 . . . . 5 ((𝑅𝑉 ∧ 1 ∈ ℕ) → (𝑅𝑟(1 + 1)) = ((𝑅𝑟1) ∘ 𝑅))
2927, 28mpan2 691 . . . 4 (𝑅𝑉 → (𝑅𝑟(1 + 1)) = ((𝑅𝑟1) ∘ 𝑅))
3025coeq2d 5876 . . . 4 (𝑅𝑉 → (𝑅 ∘ (𝑅𝑟1)) = (𝑅𝑅))
3126, 29, 303eqtr4d 2785 . . 3 (𝑅𝑉 → (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1)))
32 coeq1 5871 . . . . . . . . 9 ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = ((𝑅 ∘ (𝑅𝑟𝑚)) ∘ 𝑅))
33 coass 6287 . . . . . . . . 9 ((𝑅 ∘ (𝑅𝑟𝑚)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅))
3432, 33eqtrdi 2791 . . . . . . . 8 ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
3534adantl 481 . . . . . . 7 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
36 simpl 482 . . . . . . . 8 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑉𝑚 ∈ ℕ))
37 peano2nn 12276 . . . . . . . . 9 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
3837anim2i 617 . . . . . . . 8 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑉 ∧ (𝑚 + 1) ∈ ℕ))
39 relexpsucnnr 15061 . . . . . . . 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 15061 . . . . . . . . 9 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
4241adantr 480 . . . . . . . 8 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
4342coeq2d 5876 . . . . . . 7 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅 ∘ (𝑅𝑟(𝑚 + 1))) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
4435, 40, 433eqtr4d 2785 . . . . . 6 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))
4544ex 412 . . . . 5 ((𝑅𝑉𝑚 ∈ ℕ) → ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1)))))
4645expcom 413 . . . 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 12282 . 2 (𝑁 ∈ ℕ → (𝑅𝑉 → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁))))
4948impcom 407 1 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  ccom 5693  (class class class)co 7431  1c1 11154   + caddc 11156  cn 12264  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:  relexpsucl  15067  relexpcnv  15071  relexpaddnn  15087  trclfvcom  43713  trclimalb2  43716
  Copyright terms: Public domain W3C validator