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Theorem relexpcnv 14152
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 11620 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 oveq2 6913 . . . . . . . 8 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
32cnveqd 5530 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
4 oveq2 6913 . . . . . . 7 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
53, 4eqeq12d 2840 . . . . . 6 (𝑛 = 1 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟1) = (𝑅𝑟1)))
65imbi2d 332 . . . . 5 (𝑛 = 1 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟1) = (𝑅𝑟1))))
7 oveq2 6913 . . . . . . . 8 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
87cnveqd 5530 . . . . . . 7 (𝑛 = 𝑚(𝑅𝑟𝑛) = (𝑅𝑟𝑚))
9 oveq2 6913 . . . . . . 7 (𝑛 = 𝑚 → (𝑅𝑟𝑛) = (𝑅𝑟𝑚))
108, 9eqeq12d 2840 . . . . . 6 (𝑛 = 𝑚 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟𝑚) = (𝑅𝑟𝑚)))
1110imbi2d 332 . . . . 5 (𝑛 = 𝑚 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚))))
12 oveq2 6913 . . . . . . . 8 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1312cnveqd 5530 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
14 oveq2 6913 . . . . . . 7 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
1513, 14eqeq12d 2840 . . . . . 6 (𝑛 = (𝑚 + 1) → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1))))
1615imbi2d 332 . . . . 5 (𝑛 = (𝑚 + 1) → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1)))))
17 oveq2 6913 . . . . . . . 8 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
1817cnveqd 5530 . . . . . . 7 (𝑛 = 𝑁(𝑅𝑟𝑛) = (𝑅𝑟𝑁))
19 oveq2 6913 . . . . . . 7 (𝑛 = 𝑁 → (𝑅𝑟𝑛) = (𝑅𝑟𝑁))
2018, 19eqeq12d 2840 . . . . . 6 (𝑛 = 𝑁 → ((𝑅𝑟𝑛) = (𝑅𝑟𝑛) ↔ (𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
2120imbi2d 332 . . . . 5 (𝑛 = 𝑁 → ((𝑅𝑉(𝑅𝑟𝑛) = (𝑅𝑟𝑛)) ↔ (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁))))
22 relexp1g 14143 . . . . . . 7 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2322cnveqd 5530 . . . . . 6 (𝑅𝑉(𝑅𝑟1) = 𝑅)
24 cnvexg 7374 . . . . . . 7 (𝑅𝑉𝑅 ∈ V)
25 relexp1g 14143 . . . . . . 7 (𝑅 ∈ V → (𝑅𝑟1) = 𝑅)
2624, 25syl 17 . . . . . 6 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
2723, 26eqtr4d 2864 . . . . 5 (𝑅𝑉(𝑅𝑟1) = (𝑅𝑟1))
28 cnvco 5540 . . . . . . . . 9 ((𝑅𝑟𝑚) ∘ 𝑅) = (𝑅(𝑅𝑟𝑚))
29 simp3 1172 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟𝑚) = (𝑅𝑟𝑚))
3029coeq2d 5517 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅(𝑅𝑟𝑚)) = (𝑅 ∘ (𝑅𝑟𝑚)))
3128, 30syl5eq 2873 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → ((𝑅𝑟𝑚) ∘ 𝑅) = (𝑅 ∘ (𝑅𝑟𝑚)))
32 simp2 1171 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑅𝑉)
33 simp1 1170 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑚 ∈ ℕ)
34 relexpsucnnr 14142 . . . . . . . . . 10 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3532, 33, 34syl2anc 579 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3635cnveqd 5530 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
3732, 24syl 17 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → 𝑅 ∈ V)
38 relexpsucnnl 14149 . . . . . . . . 9 ((𝑅 ∈ V ∧ 𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))
3937, 33, 38syl2anc 579 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = (𝑅 ∘ (𝑅𝑟𝑚)))
4031, 36, 393eqtr4d 2871 . . . . . . 7 ((𝑚 ∈ ℕ ∧ 𝑅𝑉(𝑅𝑟𝑚) = (𝑅𝑟𝑚)) → (𝑅𝑟(𝑚 + 1)) = (𝑅𝑟(𝑚 + 1)))
41403exp 1152 . . . . . 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 11370 . . . 4 (𝑁 ∈ ℕ → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
44 cnvresid 6201 . . . . . . 7 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
45 uncom 3984 . . . . . . . . 9 (dom 𝑅 ∪ ran 𝑅) = (ran 𝑅 ∪ dom 𝑅)
46 df-rn 5353 . . . . . . . . . 10 ran 𝑅 = dom 𝑅
47 dfdm4 5548 . . . . . . . . . 10 dom 𝑅 = ran 𝑅
4846, 47uneq12i 3992 . . . . . . . . 9 (ran 𝑅 ∪ dom 𝑅) = (dom 𝑅 ∪ ran 𝑅)
4945, 48eqtri 2849 . . . . . . . 8 (dom 𝑅 ∪ ran 𝑅) = (dom 𝑅 ∪ ran 𝑅)
5049reseq2i 5626 . . . . . . 7 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
5144, 50eqtri 2849 . . . . . 6 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
52 oveq2 6913 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑟𝑁) = (𝑅𝑟0))
53 relexp0g 14139 . . . . . . . 8 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5452, 53sylan9eq 2881 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5554cnveqd 5530 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
56 oveq2 6913 . . . . . . . 8 (𝑁 = 0 → (𝑅𝑟𝑁) = (𝑅𝑟0))
5756adantr 474 . . . . . . 7 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
58 simpr 479 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
59 relexp0g 14139 . . . . . . . 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 2861 . . . . . 6 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
6251, 55, 613eqtr4a 2887 . . . . 5 ((𝑁 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
6362ex 403 . . . 4 (𝑁 = 0 → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
6443, 63jaoi 888 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
651, 64sylbi 209 . 2 (𝑁 ∈ ℕ0 → (𝑅𝑉(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))
6665imp 397 1 ((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wo 878  w3a 1111   = wceq 1656  wcel 2164  Vcvv 3414  cun 3796   I cid 5249  ccnv 5341  dom cdm 5342  ran crn 5343  cres 5344  ccom 5346  (class class class)co 6905  0cc0 10252  1c1 10253   + caddc 10255  cn 11350  0cn0 11618  𝑟crelexp 14137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-n0 11619  df-z 11705  df-uz 11969  df-seq 13096  df-relexp 14138
This theorem is referenced by:  relexpcnvd  14153  relexpnnrn  14162  relexpaddg  14170  relexpaddss  38844  cnvtrclfv  38850
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