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Theorem relexpcnv 14982
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 12474 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 oveq2 7417 . . . . . . . 8 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
32cnveqd 5876 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
4 oveq2 7417 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
53, 4eqeq12d 2749 . . . . . 6 (𝑛 = 1 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟1) = (𝑅𝑟1)))
65imbi2d 341 . . . . 5 (𝑛 = 1 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟1) = (𝑅𝑟1))))
7 oveq2 7417 . . . . . . . 8 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
87cnveqd 5876 . . . . . . 7 (𝑛 = 𝑚(𝑅𝑟𝑛) = (𝑅𝑟𝑚))
9 oveq2 7417 . . . . . . 7 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
108, 9eqeq12d 2749 . . . . . 6 (𝑛 = 𝑚 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟𝑚) = (𝑅𝑟𝑚)))
1110imbi2d 341 . . . . 5 (𝑛 = 𝑚 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚))))
12 oveq2 7417 . . . . . . . 8 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1312cnveqd 5876 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
14 oveq2 7417 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1513, 14eqeq12d 2749 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1))))
1615imbi2d 341 . . . . 5 (𝑛 = (𝑚 + 1) → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1)))))
17 oveq2 7417 . . . . . . . 8 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
1817cnveqd 5876 . . . . . . 7 (𝑛 = 𝑁(𝑅𝑟𝑛) = (𝑅𝑟𝑁))
19 oveq2 7417 . . . . . . 7 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2018, 19eqeq12d 2749 . . . . . 6 (𝑛 = 𝑁 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
2120imbi2d 341 . . . . 5 (𝑛 = 𝑁 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁))))
22 relexp1g 14973 . . . . . . 7 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2322cnveqd 5876 . . . . . 6 (𝑅𝑉(𝑅𝑟1) = 𝑅)
24 cnvexg 7915 . . . . . . 7 (𝑅𝑉𝑅 ∈ V)
25 relexp1g 14973 . . . . . . 7 (𝑅 ∈ V → (𝑅𝑟1) = 𝑅)
2624, 25syl 17 . . . . . 6 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2723, 26eqtr4d 2776 . . . . 5 (𝑅𝑉(𝑅𝑟1) = (𝑅𝑟1))
28 cnvco 5886 . . . . . . . . 9 ((𝑅𝑟𝑚) ∘ 𝑅) = (𝑅(𝑅𝑟𝑚))
29 simp3 1139 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟𝑚) = (𝑅𝑟𝑚))
3029coeq2d 5863 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅(𝑅𝑟𝑚)) = (𝑅 ∘ (𝑅𝑟𝑚)))
3128, 30eqtrid 2785 . . . . . . . 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 14972 . . . . . . . . . 10 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3532, 33, 34syl2anc 585 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3635cnveqd 5876 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3732, 24syl 17 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑅 ∈ V)
38 relexpsucnnl 14977 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))
3937, 33, 38syl2anc 585 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))
4031, 36, 393eqtr4d 2783 . . . . . . 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 12230 . . . 4 (𝑁 ∈ ℕ → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
44 cnvresid 6628 . . . . . . 7 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
45 uncom 4154 . . . . . . . . 9 (dom 𝑅 ∪ ran 𝑅) = (ran 𝑅 ∪ dom 𝑅)
46 df-rn 5688 . . . . . . . . . 10 ran 𝑅 = dom 𝑅
47 dfdm4 5896 . . . . . . . . . 10 dom 𝑅 = ran 𝑅
4846, 47uneq12i 4162 . . . . . . . . 9 (ran 𝑅 ∪ dom 𝑅) = (dom 𝑅 ∪ ran 𝑅)
4945, 48eqtri 2761 . . . . . . . 8 (dom 𝑅 ∪ ran 𝑅) = (dom 𝑅 ∪ ran 𝑅)
5049reseq2i 5979 . . . . . . 7 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
5144, 50eqtri 2761 . . . . . 6 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
52 oveq2 7417 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑟𝑁) = (𝑅𝑟0))
53 relexp0g 14969 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5452, 53sylan9eq 2793 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5554cnveqd 5876 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
56 oveq2 7417 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑟𝑁) = (𝑅𝑟0))
5756adantr 482 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
58 simpr 486 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
59 relexp0g 14969 . . . . . . . 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 2773 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6251, 55, 613eqtr4a 2799 . . . . 5 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
6362ex 414 . . . 4 (𝑁 = 0 → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
6443, 63jaoi 856 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
651, 64sylbi 216 . 2 (𝑁 ∈ ℕ0 → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
6665imp 408 1 ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  cun 3947   I cid 5574  ccnv 5676  dom cdm 5677  ran crn 5678  cres 5679  ccom 5681  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113  cn 12212  0cn0 12472  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:  relexpcnvd  14983  relexpnnrn  14992  relexpaddg  15000  cnvtrclfv  42475
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