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Theorem cshweqdif2 13416
 Description: If cyclically shifting two words (of the same length) results in the same word, cyclically shifting one of the words by the difference of the numbers of shifts results in the other word. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 6-Jun-2018.) (Revised by AV, 1-Nov-2018.)
Assertion
Ref Expression
cshweqdif2 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀) → (𝑈 cyclShift (𝑀𝑁)) = 𝑊))

Proof of Theorem cshweqdif2
StepHypRef Expression
1 simpr 476 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) → 𝑈 ∈ Word 𝑉)
21adantr 480 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑈 ∈ Word 𝑉)
3 zsubcl 11296 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁) ∈ ℤ)
43ancoms 468 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑀𝑁) ∈ ℤ)
54adantl 481 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑀𝑁) ∈ ℤ)
6 simpr 476 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
76adantl 481 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑀 ∈ ℤ)
82, 5, 73jca 1235 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑈 ∈ Word 𝑉 ∧ (𝑀𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ))
98adantr 480 . . . . . 6 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 ∈ Word 𝑉 ∧ (𝑀𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ))
10 3cshw 13415 . . . . . 6 ((𝑈 ∈ Word 𝑉 ∧ (𝑀𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑈 cyclShift (𝑀𝑁)) = (((𝑈 cyclShift 𝑀) cyclShift (𝑀𝑁)) cyclShift ((#‘𝑈) − 𝑀)))
119, 10syl 17 . . . . 5 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 cyclShift (𝑀𝑁)) = (((𝑈 cyclShift 𝑀) cyclShift (𝑀𝑁)) cyclShift ((#‘𝑈) − 𝑀)))
12 simpl 472 . . . . . . . . . 10 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉))
1312ancomd 466 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑈 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
1413adantr 480 . . . . . . . 8 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
15 simpr 476 . . . . . . . . . 10 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
1615ancomd 466 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
1716adantr 480 . . . . . . . 8 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
18 simpr 476 . . . . . . . . 9 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀))
1918eqcomd 2616 . . . . . . . 8 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 cyclShift 𝑀) = (𝑊 cyclShift 𝑁))
20 cshwleneq 13414 . . . . . . . 8 (((𝑈 ∈ Word 𝑉𝑊 ∈ Word 𝑉) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑈 cyclShift 𝑀) = (𝑊 cyclShift 𝑁)) → (#‘𝑈) = (#‘𝑊))
2114, 17, 19, 20syl3anc 1318 . . . . . . 7 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (#‘𝑈) = (#‘𝑊))
2221oveq1d 6564 . . . . . 6 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → ((#‘𝑈) − 𝑀) = ((#‘𝑊) − 𝑀))
2322oveq2d 6565 . . . . 5 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (((𝑈 cyclShift 𝑀) cyclShift (𝑀𝑁)) cyclShift ((#‘𝑈) − 𝑀)) = (((𝑈 cyclShift 𝑀) cyclShift (𝑀𝑁)) cyclShift ((#‘𝑊) − 𝑀)))
2411, 23eqtrd 2644 . . . 4 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 cyclShift (𝑀𝑁)) = (((𝑈 cyclShift 𝑀) cyclShift (𝑀𝑁)) cyclShift ((#‘𝑊) − 𝑀)))
2519oveq1d 6564 . . . . . 6 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → ((𝑈 cyclShift 𝑀) cyclShift (𝑀𝑁)) = ((𝑊 cyclShift 𝑁) cyclShift (𝑀𝑁)))
26 simpl 472 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) → 𝑊 ∈ Word 𝑉)
2726adantr 480 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑊 ∈ Word 𝑉)
28 simpl 472 . . . . . . . . . 10 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑁 ∈ ℤ)
2928adantl 481 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑁 ∈ ℤ)
3027, 29, 53jca 1235 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ (𝑀𝑁) ∈ ℤ))
3130adantr 480 . . . . . . 7 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ (𝑀𝑁) ∈ ℤ))
32 2cshw 13410 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ (𝑀𝑁) ∈ ℤ) → ((𝑊 cyclShift 𝑁) cyclShift (𝑀𝑁)) = (𝑊 cyclShift (𝑁 + (𝑀𝑁))))
3331, 32syl 17 . . . . . 6 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → ((𝑊 cyclShift 𝑁) cyclShift (𝑀𝑁)) = (𝑊 cyclShift (𝑁 + (𝑀𝑁))))
34 zcn 11259 . . . . . . . . . . 11 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
35 zcn 11259 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
3634, 35anim12i 588 . . . . . . . . . 10 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ))
3736adantl 481 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ))
3837adantr 480 . . . . . . . 8 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ))
39 pncan3 10168 . . . . . . . 8 ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 + (𝑀𝑁)) = 𝑀)
4038, 39syl 17 . . . . . . 7 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑁 + (𝑀𝑁)) = 𝑀)
4140oveq2d 6565 . . . . . 6 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 cyclShift (𝑁 + (𝑀𝑁))) = (𝑊 cyclShift 𝑀))
4225, 33, 413eqtrd 2648 . . . . 5 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → ((𝑈 cyclShift 𝑀) cyclShift (𝑀𝑁)) = (𝑊 cyclShift 𝑀))
4342oveq1d 6564 . . . 4 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (((𝑈 cyclShift 𝑀) cyclShift (𝑀𝑁)) cyclShift ((#‘𝑊) − 𝑀)) = ((𝑊 cyclShift 𝑀) cyclShift ((#‘𝑊) − 𝑀)))
44 lencl 13179 . . . . . . . . . 10 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
4544nn0zd 11356 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℤ)
4645adantr 480 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) → (#‘𝑊) ∈ ℤ)
47 zsubcl 11296 . . . . . . . 8 (((#‘𝑊) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((#‘𝑊) − 𝑀) ∈ ℤ)
4846, 6, 47syl2an 493 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((#‘𝑊) − 𝑀) ∈ ℤ)
4927, 7, 483jca 1235 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ ((#‘𝑊) − 𝑀) ∈ ℤ))
5049adantr 480 . . . . 5 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ ((#‘𝑊) − 𝑀) ∈ ℤ))
51 2cshw 13410 . . . . 5 ((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ ((#‘𝑊) − 𝑀) ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift ((#‘𝑊) − 𝑀)) = (𝑊 cyclShift (𝑀 + ((#‘𝑊) − 𝑀))))
5250, 51syl 17 . . . 4 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → ((𝑊 cyclShift 𝑀) cyclShift ((#‘𝑊) − 𝑀)) = (𝑊 cyclShift (𝑀 + ((#‘𝑊) − 𝑀))))
5324, 43, 523eqtrd 2648 . . 3 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 cyclShift (𝑀𝑁)) = (𝑊 cyclShift (𝑀 + ((#‘𝑊) − 𝑀))))
5444nn0cnd 11230 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℂ)
5554adantr 480 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) → (#‘𝑊) ∈ ℂ)
5635adantl 481 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℂ)
5755, 56anim12i 588 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((#‘𝑊) ∈ ℂ ∧ 𝑀 ∈ ℂ))
5857ancomd 466 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑀 ∈ ℂ ∧ (#‘𝑊) ∈ ℂ))
5958adantr 480 . . . . 5 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑀 ∈ ℂ ∧ (#‘𝑊) ∈ ℂ))
60 pncan3 10168 . . . . 5 ((𝑀 ∈ ℂ ∧ (#‘𝑊) ∈ ℂ) → (𝑀 + ((#‘𝑊) − 𝑀)) = (#‘𝑊))
6159, 60syl 17 . . . 4 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑀 + ((#‘𝑊) − 𝑀)) = (#‘𝑊))
6261oveq2d 6565 . . 3 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 cyclShift (𝑀 + ((#‘𝑊) − 𝑀))) = (𝑊 cyclShift (#‘𝑊)))
63 cshwn 13394 . . . . 5 (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (#‘𝑊)) = 𝑊)
6427, 63syl 17 . . . 4 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊 cyclShift (#‘𝑊)) = 𝑊)
6564adantr 480 . . 3 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑊 cyclShift (#‘𝑊)) = 𝑊)
6653, 62, 653eqtrd 2648 . 2 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (𝑈 cyclShift (𝑀𝑁)) = 𝑊)
6766ex 449 1 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀) → (𝑈 cyclShift (𝑀𝑁)) = 𝑊))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   + caddc 9818   − cmin 10145  ℤcz 11254  #chash 12979  Word cword 13146   cyclShift ccsh 13385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386 This theorem is referenced by:  cshweqdifid  13417
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