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Theorem relexpsucnnl 13701
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 6612 . . . . . 6 (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
21oveq2d 6621 . . . . 5 (𝑛 = 1 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(1 + 1)))
3 oveq2 6613 . . . . . 6 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
43coeq2d 5249 . . . . 5 (𝑛 = 1 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟1)))
52, 4eqeq12d 2641 . . . 4 (𝑛 = 1 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1))))
65imbi2d 330 . . 3 (𝑛 = 1 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1)))))
7 oveq1 6612 . . . . . 6 (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
87oveq2d 6621 . . . . 5 (𝑛 = 𝑚 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(𝑚 + 1)))
9 oveq2 6613 . . . . . 6 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
109coeq2d 5249 . . . . 5 (𝑛 = 𝑚 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟𝑚)))
118, 10eqeq12d 2641 . . . 4 (𝑛 = 𝑚 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))))
1211imbi2d 330 . . 3 (𝑛 = 𝑚 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))))
13 oveq1 6612 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑛 + 1) = ((𝑚 + 1) + 1))
1413oveq2d 6621 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟((𝑚 + 1) + 1)))
15 oveq2 6613 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1615coeq2d 5249 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))
1714, 16eqeq12d 2641 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1)))))
1817imbi2d 330 . . 3 (𝑛 = (𝑚 + 1) → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))))
19 oveq1 6612 . . . . . 6 (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
2019oveq2d 6621 . . . . 5 (𝑛 = 𝑁 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(𝑁 + 1)))
21 oveq2 6613 . . . . . 6 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2221coeq2d 5249 . . . . 5 (𝑛 = 𝑁 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟𝑁)))
2320, 22eqeq12d 2641 . . . 4 (𝑛 = 𝑁 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁))))
2423imbi2d 330 . . 3 (𝑛 = 𝑁 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))))
25 relexp1g 13695 . . . . 5 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2625coeq1d 5248 . . . 4 (𝑅𝑉 → ((𝑅𝑟1) ∘ 𝑅) = (𝑅𝑅))
27 1nn 10976 . . . . 5 1 ∈ ℕ
28 relexpsucnnr 13694 . . . . 5 ((𝑅𝑉 ∧ 1 ∈ ℕ) → (𝑅𝑟(1 + 1)) = ((𝑅𝑟1) ∘ 𝑅))
2927, 28mpan2 706 . . . 4 (𝑅𝑉 → (𝑅𝑟(1 + 1)) = ((𝑅𝑟1) ∘ 𝑅))
3025coeq2d 5249 . . . 4 (𝑅𝑉 → (𝑅 ∘ (𝑅𝑟1)) = (𝑅𝑅))
3126, 29, 303eqtr4d 2670 . . 3 (𝑅𝑉 → (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1)))
32 coeq1 5244 . . . . . . . . 9 ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = ((𝑅 ∘ (𝑅𝑟𝑚)) ∘ 𝑅))
33 coass 5616 . . . . . . . . 9 ((𝑅 ∘ (𝑅𝑟𝑚)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅))
3432, 33syl6eq 2676 . . . . . . . 8 ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
3534adantl 482 . . . . . . 7 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
36 simpl 473 . . . . . . . 8 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑉𝑚 ∈ ℕ))
37 peano2nn 10977 . . . . . . . . 9 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
3837anim2i 592 . . . . . . . 8 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑉 ∧ (𝑚 + 1) ∈ ℕ))
39 relexpsucnnr 13694 . . . . . . . 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 13694 . . . . . . . . 9 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
4241adantr 481 . . . . . . . 8 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
4342coeq2d 5249 . . . . . . 7 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅 ∘ (𝑅𝑟(𝑚 + 1))) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
4435, 40, 433eqtr4d 2670 . . . . . 6 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))
4544ex 450 . . . . 5 ((𝑅𝑉𝑚 ∈ ℕ) → ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1)))))
4645expcom 451 . . . 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 10983 . 2 (𝑁 ∈ ℕ → (𝑅𝑉 → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁))))
4948impcom 446 1 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  ccom 5083  (class class class)co 6605  1c1 9882   + caddc 9884  cn 10965  𝑟crelexp 13689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-seq 12739  df-relexp 13690
This theorem is referenced by:  relexpsucl  13702  relexpcnv  13704  relexpaddnn  13720  trclfvcom  37482  trclimalb2  37485
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