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Theorem relexpcnv 15074
Description: Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.)
Assertion
Ref Expression
relexpcnv ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))

Proof of Theorem relexpcnv
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn0 12528 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 oveq2 7439 . . . . . . . 8 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
32cnveqd 5886 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
4 oveq2 7439 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
53, 4eqeq12d 2753 . . . . . 6 (𝑛 = 1 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟1) = (𝑅𝑟1)))
65imbi2d 340 . . . . 5 (𝑛 = 1 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟1) = (𝑅𝑟1))))
7 oveq2 7439 . . . . . . . 8 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
87cnveqd 5886 . . . . . . 7 (𝑛 = 𝑚(𝑅𝑟𝑛) = (𝑅𝑟𝑚))
9 oveq2 7439 . . . . . . 7 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
108, 9eqeq12d 2753 . . . . . 6 (𝑛 = 𝑚 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟𝑚) = (𝑅𝑟𝑚)))
1110imbi2d 340 . . . . 5 (𝑛 = 𝑚 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚))))
12 oveq2 7439 . . . . . . . 8 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1312cnveqd 5886 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
14 oveq2 7439 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1513, 14eqeq12d 2753 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1))))
1615imbi2d 340 . . . . 5 (𝑛 = (𝑚 + 1) → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1)))))
17 oveq2 7439 . . . . . . . 8 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
1817cnveqd 5886 . . . . . . 7 (𝑛 = 𝑁(𝑅𝑟𝑛) = (𝑅𝑟𝑁))
19 oveq2 7439 . . . . . . 7 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2018, 19eqeq12d 2753 . . . . . 6 (𝑛 = 𝑁 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
2120imbi2d 340 . . . . 5 (𝑛 = 𝑁 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁))))
22 relexp1g 15065 . . . . . . 7 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2322cnveqd 5886 . . . . . 6 (𝑅𝑉(𝑅𝑟1) = 𝑅)
24 cnvexg 7946 . . . . . . 7 (𝑅𝑉𝑅 ∈ V)
25 relexp1g 15065 . . . . . . 7 (𝑅 ∈ V → (𝑅𝑟1) = 𝑅)
2624, 25syl 17 . . . . . 6 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2723, 26eqtr4d 2780 . . . . 5 (𝑅𝑉(𝑅𝑟1) = (𝑅𝑟1))
28 cnvco 5896 . . . . . . . . 9 ((𝑅𝑟𝑚) ∘ 𝑅) = (𝑅(𝑅𝑟𝑚))
29 simp3 1139 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟𝑚) = (𝑅𝑟𝑚))
3029coeq2d 5873 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅(𝑅𝑟𝑚)) = (𝑅 ∘ (𝑅𝑟𝑚)))
3128, 30eqtrid 2789 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → ((𝑅𝑟𝑚) ∘ 𝑅) = (𝑅 ∘ (𝑅𝑟𝑚)))
32 simp2 1138 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑅𝑉)
33 simp1 1137 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑚 ∈ ℕ)
34 relexpsucnnr 15064 . . . . . . . . . 10 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3532, 33, 34syl2anc 584 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3635cnveqd 5886 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3732, 24syl 17 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑅 ∈ V)
38 relexpsucnnl 15069 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))
3937, 33, 38syl2anc 584 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))
4031, 36, 393eqtr4d 2787 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1)))
41403exp 1120 . . . . . 6 (𝑚 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑚) = (𝑅𝑟𝑚) → (𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1)))))
4241a2d 29 . . . . 5 (𝑚 ∈ ℕ → ((𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑉(𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1)))))
436, 11, 16, 21, 27, 42nnind 12284 . . . 4 (𝑁 ∈ ℕ → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
44 cnvresid 6645 . . . . . . 7 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
45 uncom 4158 . . . . . . . . 9 (dom 𝑅 ∪ ran 𝑅) = (ran 𝑅 ∪ dom 𝑅)
46 df-rn 5696 . . . . . . . . . 10 ran 𝑅 = dom 𝑅
47 dfdm4 5906 . . . . . . . . . 10 dom 𝑅 = ran 𝑅
4846, 47uneq12i 4166 . . . . . . . . 9 (ran 𝑅 ∪ dom 𝑅) = (dom 𝑅 ∪ ran 𝑅)
4945, 48eqtri 2765 . . . . . . . 8 (dom 𝑅 ∪ ran 𝑅) = (dom 𝑅 ∪ ran 𝑅)
5049reseq2i 5994 . . . . . . 7 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
5144, 50eqtri 2765 . . . . . 6 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
52 oveq2 7439 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑟𝑁) = (𝑅𝑟0))
53 relexp0g 15061 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5452, 53sylan9eq 2797 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5554cnveqd 5886 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
56 oveq2 7439 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑟𝑁) = (𝑅𝑟0))
5756adantr 480 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
58 simpr 484 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
59 relexp0g 15061 . . . . . . . 8 (𝑅 ∈ V → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6058, 24, 593syl 18 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6157, 60eqtrd 2777 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6251, 55, 613eqtr4a 2803 . . . . 5 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
6362ex 412 . . . 4 (𝑁 = 0 → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
6443, 63jaoi 858 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
651, 64sylbi 217 . 2 (𝑁 ∈ ℕ0 → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
6665imp 406 1 ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949   I cid 5577  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  ccom 5689  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158  cn 12266  0cn0 12526  𝑟crelexp 15058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-relexp 15059
This theorem is referenced by:  relexpcnvd  15075  relexpnnrn  15084  relexpaddg  15092  cnvtrclfv  43737
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