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Theorem relexpmulg 41042
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 12121 . . . 4 (𝐽 ∈ ℕ0 ↔ (𝐽 ∈ ℕ ∨ 𝐽 = 0))
2 elnn0 12121 . . . . . 6 (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
3 relexpmulnn 41041 . . . . . . . . . 10 (((𝑅𝑉𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
433adantl3 1170 . . . . . . . . 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 773 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐼 = (𝐽 · 𝐾))
8 simpll 767 . . . . . . . . . . . . . 14 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐾 = 0)
98oveq2d 7250 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → (𝐽 · 𝐾) = (𝐽 · 0))
10 simplr 769 . . . . . . . . . . . . . . 15 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℕ)
1110nncnd 11875 . . . . . . . . . . . . . 14 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → 𝐽 ∈ ℂ)
1211mul01d 11060 . . . . . . . . . . . . 13 (((𝐾 = 0 ∧ 𝐽 ∈ ℕ) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾))) → (𝐽 · 0) = 0)
137, 9, 123eqtrd 2783 . . . . . . . . . . . 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 11892 . . . . . . . . . . . . . . 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 5081 . . . . . . . . . . . . . 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 1350 . . . . . . . 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 857 . . . . . 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 769 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐽 = 0)
2928oveq2d 7250 . . . . . . . . . 10 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝑅𝑟𝐽) = (𝑅𝑟0))
30 simpr1 1196 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝑅𝑉)
31 relexp0g 14617 . . . . . . . . . . 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 2779 . . . . . . . . 9 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝑅𝑟𝐽) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3433oveq1d 7249 . . . . . . . 8 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (( I ↾ (dom 𝑅 ∪ ran 𝑅))↑𝑟𝐾))
35 dmexg 7702 . . . . . . . . . . 11 (𝑅𝑉 → dom 𝑅 ∈ V)
36 rnexg 7703 . . . . . . . . . . 11 (𝑅𝑉 → ran 𝑅 ∈ V)
37 unexg 7555 . . . . . . . . . . 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 767 . . . . . . . . 9 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐾 ∈ ℕ0)
41 relexpiidm 41036 . . . . . . . . 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 1197 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐼 = (𝐽 · 𝐾))
4428oveq1d 7249 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝐽 · 𝐾) = (0 · 𝐾))
4540nn0cnd 12181 . . . . . . . . . . . 12 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐾 ∈ ℂ)
4645mul02d 11059 . . . . . . . . . . 11 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (0 · 𝐾) = 0)
4743, 44, 463eqtrd 2783 . . . . . . . . . 10 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → 𝐼 = 0)
4847oveq2d 7250 . . . . . . . . 9 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → (𝑅𝑟𝐼) = (𝑅𝑟0))
4948, 32eqtr2d 2780 . . . . . . . 8 (((𝐾 ∈ ℕ0𝐽 = 0) ∧ (𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝐼))
5034, 42, 493eqtrd 2783 . . . . . . 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 859 . . . 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 847  w3a 1089   = wceq 1543  wcel 2112  Vcvv 3423  cun 3881   class class class wbr 5069   I cid 5470  dom cdm 5568  ran crn 5569  cres 5570  (class class class)co 7234  0cc0 10758   · cmul 10763  cle 10897  cn 11859  0cn0 12119  𝑟crelexp 14614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544  ax-cnex 10814  ax-resscn 10815  ax-1cn 10816  ax-icn 10817  ax-addcl 10818  ax-addrcl 10819  ax-mulcl 10820  ax-mulrcl 10821  ax-mulcom 10822  ax-addass 10823  ax-mulass 10824  ax-distr 10825  ax-i2m1 10826  ax-1ne0 10827  ax-1rid 10828  ax-rnegex 10829  ax-rrecex 10830  ax-cnre 10831  ax-pre-lttri 10832  ax-pre-lttrn 10833  ax-pre-ltadd 10834  ax-pre-mulgt0 10835
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-pred 6178  df-ord 6236  df-on 6237  df-lim 6238  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-riota 7191  df-ov 7237  df-oprab 7238  df-mpo 7239  df-om 7666  df-2nd 7783  df-wrecs 8070  df-recs 8131  df-rdg 8169  df-er 8414  df-en 8650  df-dom 8651  df-sdom 8652  df-pnf 10898  df-mnf 10899  df-xr 10900  df-ltxr 10901  df-le 10902  df-sub 11093  df-neg 11094  df-nn 11860  df-n0 12120  df-z 12206  df-uz 12468  df-seq 13606  df-relexp 14615
This theorem is referenced by: (None)
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