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Theorem relexpcnv 15014
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 12504 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 oveq2 7428 . . . . . . . 8 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
32cnveqd 5878 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
4 oveq2 7428 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
53, 4eqeq12d 2744 . . . . . 6 (𝑛 = 1 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟1) = (𝑅𝑟1)))
65imbi2d 340 . . . . 5 (𝑛 = 1 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟1) = (𝑅𝑟1))))
7 oveq2 7428 . . . . . . . 8 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
87cnveqd 5878 . . . . . . 7 (𝑛 = 𝑚(𝑅𝑟𝑛) = (𝑅𝑟𝑚))
9 oveq2 7428 . . . . . . 7 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
108, 9eqeq12d 2744 . . . . . 6 (𝑛 = 𝑚 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟𝑚) = (𝑅𝑟𝑚)))
1110imbi2d 340 . . . . 5 (𝑛 = 𝑚 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚))))
12 oveq2 7428 . . . . . . . 8 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1312cnveqd 5878 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
14 oveq2 7428 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1513, 14eqeq12d 2744 . . . . . 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 7428 . . . . . . . 8 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
1817cnveqd 5878 . . . . . . 7 (𝑛 = 𝑁(𝑅𝑟𝑛) = (𝑅𝑟𝑁))
19 oveq2 7428 . . . . . . 7 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2018, 19eqeq12d 2744 . . . . . 6 (𝑛 = 𝑁 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
2120imbi2d 340 . . . . 5 (𝑛 = 𝑁 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁))))
22 relexp1g 15005 . . . . . . 7 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2322cnveqd 5878 . . . . . 6 (𝑅𝑉(𝑅𝑟1) = 𝑅)
24 cnvexg 7932 . . . . . . 7 (𝑅𝑉𝑅 ∈ V)
25 relexp1g 15005 . . . . . . 7 (𝑅 ∈ V → (𝑅𝑟1) = 𝑅)
2624, 25syl 17 . . . . . 6 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2723, 26eqtr4d 2771 . . . . 5 (𝑅𝑉(𝑅𝑟1) = (𝑅𝑟1))
28 cnvco 5888 . . . . . . . . 9 ((𝑅𝑟𝑚) ∘ 𝑅) = (𝑅(𝑅𝑟𝑚))
29 simp3 1136 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟𝑚) = (𝑅𝑟𝑚))
3029coeq2d 5865 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅(𝑅𝑟𝑚)) = (𝑅 ∘ (𝑅𝑟𝑚)))
3128, 30eqtrid 2780 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → ((𝑅𝑟𝑚) ∘ 𝑅) = (𝑅 ∘ (𝑅𝑟𝑚)))
32 simp2 1135 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑅𝑉)
33 simp1 1134 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑚 ∈ ℕ)
34 relexpsucnnr 15004 . . . . . . . . . 10 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3532, 33, 34syl2anc 583 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3635cnveqd 5878 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3732, 24syl 17 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑅 ∈ V)
38 relexpsucnnl 15009 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))
3937, 33, 38syl2anc 583 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))
4031, 36, 393eqtr4d 2778 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1)))
41403exp 1117 . . . . . 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 12260 . . . 4 (𝑁 ∈ ℕ → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
44 cnvresid 6632 . . . . . . 7 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
45 uncom 4152 . . . . . . . . 9 (dom 𝑅 ∪ ran 𝑅) = (ran 𝑅 ∪ dom 𝑅)
46 df-rn 5689 . . . . . . . . . 10 ran 𝑅 = dom 𝑅
47 dfdm4 5898 . . . . . . . . . 10 dom 𝑅 = ran 𝑅
4846, 47uneq12i 4160 . . . . . . . . 9 (ran 𝑅 ∪ dom 𝑅) = (dom 𝑅 ∪ ran 𝑅)
4945, 48eqtri 2756 . . . . . . . 8 (dom 𝑅 ∪ ran 𝑅) = (dom 𝑅 ∪ ran 𝑅)
5049reseq2i 5982 . . . . . . 7 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
5144, 50eqtri 2756 . . . . . 6 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
52 oveq2 7428 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑟𝑁) = (𝑅𝑟0))
53 relexp0g 15001 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5452, 53sylan9eq 2788 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5554cnveqd 5878 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
56 oveq2 7428 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑟𝑁) = (𝑅𝑟0))
5756adantr 480 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
58 simpr 484 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
59 relexp0g 15001 . . . . . . . 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 2768 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6251, 55, 613eqtr4a 2794 . . . . 5 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
6362ex 412 . . . 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 406 1 ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  Vcvv 3471  cun 3945   I cid 5575  ccnv 5677  dom cdm 5678  ran crn 5679  cres 5680  ccom 5682  (class class class)co 7420  0cc0 11138  1c1 11139   + caddc 11141  cn 12242  0cn0 12502  𝑟crelexp 14998
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-er 8724  df-en 8964  df-dom 8965  df-sdom 8966  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-n0 12503  df-z 12589  df-uz 12853  df-seq 13999  df-relexp 14999
This theorem is referenced by:  relexpcnvd  15015  relexpnnrn  15024  relexpaddg  15032  cnvtrclfv  43154
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