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Theorem relexpsucnnl 14383
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 7155 . . . . . 6 (𝑛 = 1 → (𝑛 + 1) = (1 + 1))
21oveq2d 7164 . . . . 5 (𝑛 = 1 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(1 + 1)))
3 oveq2 7156 . . . . . 6 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
43coeq2d 5726 . . . . 5 (𝑛 = 1 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟1)))
52, 4eqeq12d 2835 . . . 4 (𝑛 = 1 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1))))
65imbi2d 343 . . 3 (𝑛 = 1 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1)))))
7 oveq1 7155 . . . . . 6 (𝑛 = 𝑚 → (𝑛 + 1) = (𝑚 + 1))
87oveq2d 7164 . . . . 5 (𝑛 = 𝑚 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(𝑚 + 1)))
9 oveq2 7156 . . . . . 6 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
109coeq2d 5726 . . . . 5 (𝑛 = 𝑚 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟𝑚)))
118, 10eqeq12d 2835 . . . 4 (𝑛 = 𝑚 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))))
1211imbi2d 343 . . 3 (𝑛 = 𝑚 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))))
13 oveq1 7155 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑛 + 1) = ((𝑚 + 1) + 1))
1413oveq2d 7164 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟((𝑚 + 1) + 1)))
15 oveq2 7156 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1615coeq2d 5726 . . . . 5 (𝑛 = (𝑚 + 1) → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))
1714, 16eqeq12d 2835 . . . 4 (𝑛 = (𝑚 + 1) → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1)))))
1817imbi2d 343 . . 3 (𝑛 = (𝑚 + 1) → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))))
19 oveq1 7155 . . . . . 6 (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
2019oveq2d 7164 . . . . 5 (𝑛 = 𝑁 → (𝑅𝑟(𝑛 + 1)) = (𝑅𝑟(𝑁 + 1)))
21 oveq2 7156 . . . . . 6 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2221coeq2d 5726 . . . . 5 (𝑛 = 𝑁 → (𝑅 ∘ (𝑅𝑟𝑛)) = (𝑅 ∘ (𝑅𝑟𝑁)))
2320, 22eqeq12d 2835 . . . 4 (𝑛 = 𝑁 → ((𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛)) ↔ (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁))))
2423imbi2d 343 . . 3 (𝑛 = 𝑁 → ((𝑅𝑉 → (𝑅𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅𝑟𝑛))) ↔ (𝑅𝑉 → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))))
25 relexp1g 14377 . . . . 5 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2625coeq1d 5725 . . . 4 (𝑅𝑉 → ((𝑅𝑟1) ∘ 𝑅) = (𝑅𝑅))
27 1nn 11641 . . . . 5 1 ∈ ℕ
28 relexpsucnnr 14376 . . . . 5 ((𝑅𝑉 ∧ 1 ∈ ℕ) → (𝑅𝑟(1 + 1)) = ((𝑅𝑟1) ∘ 𝑅))
2927, 28mpan2 689 . . . 4 (𝑅𝑉 → (𝑅𝑟(1 + 1)) = ((𝑅𝑟1) ∘ 𝑅))
3025coeq2d 5726 . . . 4 (𝑅𝑉 → (𝑅 ∘ (𝑅𝑟1)) = (𝑅𝑅))
3126, 29, 303eqtr4d 2864 . . 3 (𝑅𝑉 → (𝑅𝑟(1 + 1)) = (𝑅 ∘ (𝑅𝑟1)))
32 coeq1 5721 . . . . . . . . 9 ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = ((𝑅 ∘ (𝑅𝑟𝑚)) ∘ 𝑅))
33 coass 6111 . . . . . . . . 9 ((𝑅 ∘ (𝑅𝑟𝑚)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅))
3432, 33syl6eq 2870 . . . . . . . 8 ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
3534adantl 484 . . . . . . 7 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → ((𝑅𝑟(𝑚 + 1)) ∘ 𝑅) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
36 simpl 485 . . . . . . . 8 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑉𝑚 ∈ ℕ))
37 peano2nn 11642 . . . . . . . . 9 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
3837anim2i 618 . . . . . . . 8 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑉 ∧ (𝑚 + 1) ∈ ℕ))
39 relexpsucnnr 14376 . . . . . . . 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 14376 . . . . . . . . 9 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
4241adantr 483 . . . . . . . 8 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
4342coeq2d 5726 . . . . . . 7 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅 ∘ (𝑅𝑟(𝑚 + 1))) = (𝑅 ∘ ((𝑅𝑟𝑚) ∘ 𝑅)))
4435, 40, 433eqtr4d 2864 . . . . . 6 (((𝑅𝑉𝑚 ∈ ℕ) ∧ (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚))) → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1))))
4544ex 415 . . . . 5 ((𝑅𝑉𝑚 ∈ ℕ) → ((𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)) → (𝑅𝑟((𝑚 + 1) + 1)) = (𝑅 ∘ (𝑅𝑟(𝑚 + 1)))))
4645expcom 416 . . . 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 11648 . 2 (𝑁 ∈ ℕ → (𝑅𝑉 → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁))))
4948impcom 410 1 ((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  ccom 5552  (class class class)co 7148  1c1 10530   + caddc 10532  cn 11630  𝑟crelexp 14371
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-seq 13362  df-relexp 14372
This theorem is referenced by:  relexpsucl  14384  relexpcnv  14386  relexpaddnn  14402  trclfvcom  40058  trclimalb2  40061
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