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Theorem relexpmulg 40341
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 11887 . . . 4 (𝐽 ∈ ℕ0 ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
2 elnn0 11887 . . . . . 6 (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
3 relexpmulnn 40340 . . . . . . . . . 10 (((𝑅𝑉𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
433adantl3 1165 . . . . . . . . 9 (((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
54expcom 417 . . . . . . . 8 ((𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
65expcom 417 . . . . . . 7 (𝐾 ∈ ℕ → (𝐽 ∈ ℕ → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
7 simprr 772 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐼 = (𝐽 · 𝐾))
8 simpll 766 . . . . . . . . . . . . . 14 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐾 = 0)
98oveq2d 7156 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → (𝐽 · 𝐾) = (𝐽 · 0))
10 simplr 768 . . . . . . . . . . . . . . 15 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℕ)
1110nncnd 11641 . . . . . . . . . . . . . 14 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℂ)
1211mul01d 10828 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → (𝐽 · 0) = 0)
137, 9, 123eqtrd 2861 . . . . . . . . . . . 12 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐼 = 0)
14 simpl 486 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → (𝐾 = 0 ∧ 𝐽 ∈ ℕ))
15 nnnle0 11658 . . . . . . . . . . . . . . 15 (𝐽 ∈ ℕ → ¬ 𝐽 ≤ 0)
1615adantl 485 . . . . . . . . . . . . . 14 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ¬ 𝐽 ≤ 0)
17 simpl 486 . . . . . . . . . . . . . . 15 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → 𝐾 = 0)
1817breq2d 5054 . . . . . . . . . . . . . 14 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → (𝐽𝐾𝐽 ≤ 0))
1916, 18mtbird 328 . . . . . . . . . . . . 13 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ¬ 𝐽𝐾)
2014, 19syl 17 . . . . . . . . . . . 12 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → ¬ 𝐽𝐾)
2113, 20jcnd 166 . . . . . . . . . . 11 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → ¬ (𝐼 = 0 → 𝐽𝐾))
2221pm2.21d 121 . . . . . . . . . 10 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → ((𝐼 = 0 → 𝐽𝐾) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
2322exp32 424 . . . . . . . . 9 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → (𝑅𝑉 → (𝐼 = (𝐽 · 𝐾) → ((𝐼 = 0 → 𝐽𝐾) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))))
24233impd 1345 . . . . . . . 8 ((𝐾 = 0 ∧ 𝐽 ∈ ℕ) → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
2524ex 416 . . . . . . 7 (𝐾 = 0 → (𝐽 ∈ ℕ → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
266, 25jaoi 854 . . . . . 6 ((𝐾 ∈ ℕ ∨ 𝐾 = 0) → (𝐽 ∈ ℕ → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
272, 26sylbi 220 . . . . 5 (𝐾 ∈ ℕ0 → (𝐽 ∈ ℕ → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
28 simplr 768 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐽 = 0)
2928oveq2d 7156 . . . . . . . . . 10 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
30 simpr1 1191 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝑅𝑉)
31 relexp0g 14372 . . . . . . . . . . 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 2857 . . . . . . . . 9 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝑅𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3433oveq1d 7155 . . . . . . . 8 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟𝐾))
35 dmexg 7599 . . . . . . . . . . 11 (𝑅𝑉 → dom 𝑅 ∈ V)
36 rnexg 7600 . . . . . . . . . . 11 (𝑅𝑉 → ran 𝑅 ∈ V)
37 unexg 7457 . . . . . . . . . . 11 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 ∪ ran 𝑅) ∈ V)
3835, 36, 37syl2anc 587 . . . . . . . . . 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 766 . . . . . . . . 9 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐾 ∈ ℕ0)
41 relexpiidm 40335 . . . . . . . . 9 (((dom 𝑅 ∪ ran 𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟𝐾) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
4239, 40, 41syl2anc 587 . . . . . . . 8 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟𝐾) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
43 simpr2 1192 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐼 = (𝐽 · 𝐾))
4428oveq1d 7155 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝐽 · 𝐾) = (0 · 𝐾))
4540nn0cnd 11945 . . . . . . . . . . . 12 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐾 ∈ ℂ)
4645mul02d 10827 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (0 · 𝐾) = 0)
4743, 44, 463eqtrd 2861 . . . . . . . . . 10 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐼 = 0)
4847oveq2d 7156 . . . . . . . . 9 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
4948, 32eqtr2d 2858 . . . . . . . 8 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝐼))
5034, 42, 493eqtrd 2861 . . . . . . 7 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
5150ex 416 . . . . . 6 ((𝐾 ∈ ℕ0𝐽 = 0) → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
5251ex 416 . . . . 5 (𝐾 ∈ ℕ0 → (𝐽 = 0 → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
5327, 52jaod 856 . . . 4 (𝐾 ∈ ℕ0 → ((𝐽 ∈ ℕ ∨ 𝐽 = 0) → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
541, 53syl5bi 245 . . 3 (𝐾 ∈ ℕ0 → (𝐽 ∈ ℕ0 → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))))
5554impcom 411 . 2 ((𝐽 ∈ ℕ0𝐾 ∈ ℕ0) → ((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼)))
5655impcom 411 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 399  wo 844  w3a 1084   = wceq 1538  wcel 2114  Vcvv 3469  cun 3906   class class class wbr 5042   I cid 5436  dom cdm 5532  ran crn 5533  cres 5534  (class class class)co 7140  0cc0 10526   · cmul 10531  cle 10665  cn 11625  0cn0 11885  𝑟crelexp 14370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13365  df-relexp 14371
This theorem is referenced by: (None)
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