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Theorem relexpmulg 39424
Description: With ordered exponents, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
Assertion
Ref Expression
relexpmulg (((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) ∧ (𝐽 ∈ ℕ0𝐾 ∈ ℕ0)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Proof of Theorem relexpmulg
StepHypRef Expression
1 elnn0 11709 . . . 4 (𝐽 ∈ ℕ0 ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
2 elnn0 11709 . . . . . 6 (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
3 relexpmulnn 39423 . . . . . . . . . 10 (((𝑅𝑉𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
433adantl3 1148 . . . . . . . . 9 (((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
54expcom 406 . . . . . . . 8 ((𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
65expcom 406 . . . . . . 7 (𝐾 ∈ ℕ → (𝐽 ∈ ℕ → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
7 simprr 760 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐼 = (𝐽 · 𝐾))
8 simpll 754 . . . . . . . . . . . . . 14 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐾 = 0)
98oveq2d 6992 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → (𝐽 · 𝐾) = (𝐽 · 0))
10 simplr 756 . . . . . . . . . . . . . . 15 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℕ)
1110nncnd 11457 . . . . . . . . . . . . . 14 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℂ)
1211mul01d 10639 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → (𝐽 · 0) = 0)
137, 9, 123eqtrd 2818 . . . . . . . . . . . 12 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐼 = 0)
14 simpl 475 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → (𝐾 = 0 ∧ 𝐽 ∈ ℕ))
15 nnnle0 11473 . . . . . . . . . . . . . . 15 (𝐽 ∈ ℕ → ¬ 𝐽 ≤ 0)
1615adantl 474 . . . . . . . . . . . . . 14 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ¬ 𝐽 ≤ 0)
17 simpl 475 . . . . . . . . . . . . . . 15 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → 𝐾 = 0)
1817breq2d 4941 . . . . . . . . . . . . . 14 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → (𝐽𝐾𝐽 ≤ 0))
1916, 18mtbird 317 . . . . . . . . . . . . 13 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ¬ 𝐽𝐾)
2014, 19syl 17 . . . . . . . . . . . 12 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → ¬ 𝐽𝐾)
2113, 20jcn 329 . . . . . . . . . . 11 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → ¬ (𝐼 = 0 → 𝐽𝐾))
2221pm2.21d 119 . . . . . . . . . 10 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → ((𝐼 = 0 → 𝐽𝐾) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
2322exp32 413 . . . . . . . . 9 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → (𝑅𝑉 → (𝐼 = (𝐽 · 𝐾) → ((𝐼 = 0 → 𝐽𝐾) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))))
24233impd 1328 . . . . . . . 8 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
2524ex 405 . . . . . . 7 (𝐾 = 0 → (𝐽 ∈ ℕ → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
266, 25jaoi 843 . . . . . 6 ((𝐾 ∈ ℕ ∨ 𝐾 = 0) → (𝐽 ∈ ℕ → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
272, 26sylbi 209 . . . . 5 (𝐾 ∈ ℕ0 → (𝐽 ∈ ℕ → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
28 simplr 756 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐽 = 0)
2928oveq2d 6992 . . . . . . . . . 10 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
30 simpr1 1174 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝑅𝑉)
31 relexp0g 14242 . . . . . . . . . . 11 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3230, 31syl 17 . . . . . . . . . 10 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3329, 32eqtrd 2814 . . . . . . . . 9 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝑅𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3433oveq1d 6991 . . . . . . . 8 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟𝐾))
35 dmexg 7428 . . . . . . . . . . 11 (𝑅𝑉 → dom 𝑅 ∈ V)
36 rnexg 7429 . . . . . . . . . . 11 (𝑅𝑉 → ran 𝑅 ∈ V)
37 unexg 7289 . . . . . . . . . . 11 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
3835, 36, 37syl2anc 576 . . . . . . . . . 10 (𝑅𝑉 → (dom 𝑅 ∪ ran 𝑅) ∈ V)
3930, 38syl 17 . . . . . . . . 9 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
40 simpll 754 . . . . . . . . 9 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐾 ∈ ℕ0)
41 relexpiidm 39418 . . . . . . . . 9 (((dom 𝑅 ∪ ran 𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟𝐾) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
4239, 40, 41syl2anc 576 . . . . . . . 8 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟𝐾) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
43 simpr2 1175 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐼 = (𝐽 · 𝐾))
4428oveq1d 6991 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝐽 · 𝐾) = (0 · 𝐾))
4540nn0cnd 11769 . . . . . . . . . . . 12 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐾 ∈ ℂ)
4645mul02d 10638 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (0 · 𝐾) = 0)
4743, 44, 463eqtrd 2818 . . . . . . . . . 10 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐼 = 0)
4847oveq2d 6992 . . . . . . . . 9 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
4948, 32eqtr2d 2815 . . . . . . . 8 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝐼))
5034, 42, 493eqtrd 2818 . . . . . . 7 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
5150ex 405 . . . . . 6 ((𝐾 ∈ ℕ0𝐽 = 0) → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
5251ex 405 . . . . 5 (𝐾 ∈ ℕ0 → (𝐽 = 0 → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
5327, 52jaod 845 . . . 4 (𝐾 ∈ ℕ0 → ((𝐽 ∈ ℕ ∨ 𝐽 = 0) → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
541, 53syl5bi 234 . . 3 (𝐾 ∈ ℕ0 → (𝐽 ∈ ℕ0 → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
5554impcom 399 . 2 ((𝐽 ∈ ℕ0𝐾 ∈ ℕ0) → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
5655impcom 399 1 (((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) ∧ (𝐽 ∈ ℕ0𝐾 ∈ ℕ0)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  wo 833  w3a 1068   = wceq 1507  wcel 2050  Vcvv 3415  cun 3827   class class class wbr 4929   I cid 5311  dom cdm 5407  ran crn 5408  cres 5409  (class class class)co 6976  0cc0 10335   · cmul 10340  cle 10475  cn 11439  0cn0 11707  𝑟crelexp 14240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-n0 11708  df-z 11794  df-uz 12059  df-seq 13185  df-relexp 14241
This theorem is referenced by: (None)
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