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Theorem relexpcnv 13982
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 11495 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 oveq2 6800 . . . . . . . 8 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
32cnveqd 5436 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
4 oveq2 6800 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
53, 4eqeq12d 2785 . . . . . 6 (𝑛 = 1 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟1) = (𝑅𝑟1)))
65imbi2d 329 . . . . 5 (𝑛 = 1 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟1) = (𝑅𝑟1))))
7 oveq2 6800 . . . . . . . 8 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
87cnveqd 5436 . . . . . . 7 (𝑛 = 𝑚(𝑅𝑟𝑛) = (𝑅𝑟𝑚))
9 oveq2 6800 . . . . . . 7 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
108, 9eqeq12d 2785 . . . . . 6 (𝑛 = 𝑚 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟𝑚) = (𝑅𝑟𝑚)))
1110imbi2d 329 . . . . 5 (𝑛 = 𝑚 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚))))
12 oveq2 6800 . . . . . . . 8 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1312cnveqd 5436 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
14 oveq2 6800 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1513, 14eqeq12d 2785 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1))))
1615imbi2d 329 . . . . 5 (𝑛 = (𝑚 + 1) → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1)))))
17 oveq2 6800 . . . . . . . 8 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
1817cnveqd 5436 . . . . . . 7 (𝑛 = 𝑁(𝑅𝑟𝑛) = (𝑅𝑟𝑁))
19 oveq2 6800 . . . . . . 7 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2018, 19eqeq12d 2785 . . . . . 6 (𝑛 = 𝑁 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
2120imbi2d 329 . . . . 5 (𝑛 = 𝑁 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁))))
22 relexp1g 13973 . . . . . . 7 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2322cnveqd 5436 . . . . . 6 (𝑅𝑉(𝑅𝑟1) = 𝑅)
24 cnvexg 7258 . . . . . . 7 (𝑅𝑉𝑅 ∈ V)
25 relexp1g 13973 . . . . . . 7 (𝑅 ∈ V → (𝑅𝑟1) = 𝑅)
2624, 25syl 17 . . . . . 6 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2723, 26eqtr4d 2807 . . . . 5 (𝑅𝑉(𝑅𝑟1) = (𝑅𝑟1))
28 cnvco 5446 . . . . . . . . 9 ((𝑅𝑟𝑚) ∘ 𝑅) = (𝑅(𝑅𝑟𝑚))
29 simp3 1131 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟𝑚) = (𝑅𝑟𝑚))
3029coeq2d 5423 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅(𝑅𝑟𝑚)) = (𝑅 ∘ (𝑅𝑟𝑚)))
3128, 30syl5eq 2816 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → ((𝑅𝑟𝑚) ∘ 𝑅) = (𝑅 ∘ (𝑅𝑟𝑚)))
32 simp2 1130 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑅𝑉)
33 simp1 1129 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑚 ∈ ℕ)
34 relexpsucnnr 13972 . . . . . . . . . 10 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3532, 33, 34syl2anc 565 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3635cnveqd 5436 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3732, 24syl 17 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑅 ∈ V)
38 relexpsucnnl 13979 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))
3937, 33, 38syl2anc 565 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))
4031, 36, 393eqtr4d 2814 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1)))
41403exp 1111 . . . . . 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 11239 . . . 4 (𝑁 ∈ ℕ → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
44 cnvresid 6108 . . . . . . 7 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
45 uncom 3906 . . . . . . . . 9 (dom 𝑅 ∪ ran 𝑅) = (ran 𝑅 ∪ dom 𝑅)
46 df-rn 5260 . . . . . . . . . 10 ran 𝑅 = dom 𝑅
47 dfdm4 5454 . . . . . . . . . 10 dom 𝑅 = ran 𝑅
4846, 47uneq12i 3914 . . . . . . . . 9 (ran 𝑅 ∪ dom 𝑅) = (dom 𝑅 ∪ ran 𝑅)
4945, 48eqtri 2792 . . . . . . . 8 (dom 𝑅 ∪ ran 𝑅) = (dom 𝑅 ∪ ran 𝑅)
5049reseq2i 5531 . . . . . . 7 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
5144, 50eqtri 2792 . . . . . 6 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
52 oveq2 6800 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑟𝑁) = (𝑅𝑟0))
53 relexp0g 13969 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5452, 53sylan9eq 2824 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5554cnveqd 5436 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
56 oveq2 6800 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑟𝑁) = (𝑅𝑟0))
5756adantr 466 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
58 simpr 471 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
59 relexp0g 13969 . . . . . . . 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 2804 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6251, 55, 613eqtr4a 2830 . . . . 5 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
6362ex 397 . . . 4 (𝑁 = 0 → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
6443, 63jaoi 837 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
651, 64sylbi 207 . 2 (𝑁 ∈ ℕ0 → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
6665imp 393 1 ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 826  w3a 1070   = wceq 1630  wcel 2144  Vcvv 3349  cun 3719   I cid 5156  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  ccom 5253  (class class class)co 6792  0cc0 10137  1c1 10138   + caddc 10140  cn 11221  0cn0 11493  𝑟crelexp 13967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-n0 11494  df-z 11579  df-uz 11888  df-seq 13008  df-relexp 13968
This theorem is referenced by:  relexpcnvd  13983  relexpnnrn  13992  relexpaddg  14000  relexpaddss  38529  cnvtrclfv  38535
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