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Theorem clwwlkfo 26325
 Description: Lemma 4 for clwwlkbij 26327: F is an onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
Hypotheses
Ref Expression
clwwlkbij.d 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
clwwlkbij.f 𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
Assertion
Ref Expression
clwwlkfo ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → 𝐹:𝐷onto→((𝑉 ClWWalksN 𝐸)‘𝑁))
Distinct variable groups:   𝑤,𝐸   𝑤,𝑁   𝑤,𝑉   𝑡,𝐷   𝑡,𝐸,𝑤   𝑡,𝑁   𝑡,𝑉   𝑡,𝑋   𝑡,𝑌
Allowed substitution hints:   𝐷(𝑤)   𝐹(𝑤,𝑡)   𝑋(𝑤)   𝑌(𝑤)

Proof of Theorem clwwlkfo
Dummy variables 𝑖 𝑥 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwwlkbij.d . . 3 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
2 clwwlkbij.f . . 3 𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
31, 2clwwlkf 26322 . 2 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → 𝐹:𝐷⟶((𝑉 ClWWalksN 𝐸)‘𝑁))
4 clwwlknimp 26304 . . . . . . 7 (𝑝 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸))
5 simpr 476 . . . . . . . . . 10 ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) ∧ (𝑉𝑋𝐸𝑌𝑁 ∈ ℕ)) → (𝑉𝑋𝐸𝑌𝑁 ∈ ℕ))
6 simpl1 1057 . . . . . . . . . 10 ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) ∧ (𝑉𝑋𝐸𝑌𝑁 ∈ ℕ)) → (𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁))
7 3simpc 1053 . . . . . . . . . . 11 (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸))
87adantr 480 . . . . . . . . . 10 ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) ∧ (𝑉𝑋𝐸𝑌𝑁 ∈ ℕ)) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸))
91clwwlkel 26321 . . . . . . . . . 10 (((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) ∧ (𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸)) → (𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷)
105, 6, 8, 9syl3anc 1318 . . . . . . . . 9 ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) ∧ (𝑉𝑋𝐸𝑌𝑁 ∈ ℕ)) → (𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷)
11 opeq2 4341 . . . . . . . . . . . . . . 15 (𝑁 = (#‘𝑝) → ⟨0, 𝑁⟩ = ⟨0, (#‘𝑝)⟩)
1211eqcoms 2618 . . . . . . . . . . . . . 14 ((#‘𝑝) = 𝑁 → ⟨0, 𝑁⟩ = ⟨0, (#‘𝑝)⟩)
1312oveq2d 6565 . . . . . . . . . . . . 13 ((#‘𝑝) = 𝑁 → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩))
1413adantl 481 . . . . . . . . . . . 12 ((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩))
15143ad2ant1 1075 . . . . . . . . . . 11 (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩))
1615adantr 480 . . . . . . . . . 10 ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) ∧ (𝑉𝑋𝐸𝑌𝑁 ∈ ℕ)) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩))
17 simpll 786 . . . . . . . . . . . . . . . . 17 (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → 𝑝 ∈ Word 𝑉)
18 fstwrdne0 13200 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ ∧ (𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁)) → (𝑝‘0) ∈ 𝑉)
1918ancoms 468 . . . . . . . . . . . . . . . . . 18 (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑝‘0) ∈ 𝑉)
2019s1cld 13236 . . . . . . . . . . . . . . . . 17 (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → ⟨“(𝑝‘0)”⟩ ∈ Word 𝑉)
2117, 20jca 553 . . . . . . . . . . . . . . . 16 (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ 𝑁 ∈ ℕ) → (𝑝 ∈ Word 𝑉 ∧ ⟨“(𝑝‘0)”⟩ ∈ Word 𝑉))
2221ex 449 . . . . . . . . . . . . . . 15 ((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) → (𝑁 ∈ ℕ → (𝑝 ∈ Word 𝑉 ∧ ⟨“(𝑝‘0)”⟩ ∈ Word 𝑉)))
23223ad2ant1 1075 . . . . . . . . . . . . . 14 (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) → (𝑁 ∈ ℕ → (𝑝 ∈ Word 𝑉 ∧ ⟨“(𝑝‘0)”⟩ ∈ Word 𝑉)))
2423com12 32 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) → (𝑝 ∈ Word 𝑉 ∧ ⟨“(𝑝‘0)”⟩ ∈ Word 𝑉)))
25243ad2ant3 1077 . . . . . . . . . . . 12 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) → (𝑝 ∈ Word 𝑉 ∧ ⟨“(𝑝‘0)”⟩ ∈ Word 𝑉)))
2625impcom 445 . . . . . . . . . . 11 ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) ∧ (𝑉𝑋𝐸𝑌𝑁 ∈ ℕ)) → (𝑝 ∈ Word 𝑉 ∧ ⟨“(𝑝‘0)”⟩ ∈ Word 𝑉))
27 swrdccat1 13309 . . . . . . . . . . 11 ((𝑝 ∈ Word 𝑉 ∧ ⟨“(𝑝‘0)”⟩ ∈ Word 𝑉) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩) = 𝑝)
2826, 27syl 17 . . . . . . . . . 10 ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) ∧ (𝑉𝑋𝐸𝑌𝑁 ∈ ℕ)) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, (#‘𝑝)⟩) = 𝑝)
2916, 28eqtr2d 2645 . . . . . . . . 9 ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) ∧ (𝑉𝑋𝐸𝑌𝑁 ∈ ℕ)) → 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩))
3010, 29jca 553 . . . . . . . 8 ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) ∧ (𝑉𝑋𝐸𝑌𝑁 ∈ ℕ)) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩)))
3130ex 449 . . . . . . 7 (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩))))
324, 31syl 17 . . . . . 6 (𝑝 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩))))
3332impcom 445 . . . . 5 (((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) ∧ 𝑝 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩)))
34 oveq1 6556 . . . . . . 7 (𝑥 = (𝑝 ++ ⟨“(𝑝‘0)”⟩) → (𝑥 substr ⟨0, 𝑁⟩) = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩))
3534eqeq2d 2620 . . . . . 6 (𝑥 = (𝑝 ++ ⟨“(𝑝‘0)”⟩) → (𝑝 = (𝑥 substr ⟨0, 𝑁⟩) ↔ 𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩)))
3635rspcev 3282 . . . . 5 (((𝑝 ++ ⟨“(𝑝‘0)”⟩) ∈ 𝐷𝑝 = ((𝑝 ++ ⟨“(𝑝‘0)”⟩) substr ⟨0, 𝑁⟩)) → ∃𝑥𝐷 𝑝 = (𝑥 substr ⟨0, 𝑁⟩))
3733, 36syl 17 . . . 4 (((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) ∧ 𝑝 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ∃𝑥𝐷 𝑝 = (𝑥 substr ⟨0, 𝑁⟩))
381, 2clwwlkfv 26323 . . . . . . 7 (𝑥𝐷 → (𝐹𝑥) = (𝑥 substr ⟨0, 𝑁⟩))
3938eqeq2d 2620 . . . . . 6 (𝑥𝐷 → (𝑝 = (𝐹𝑥) ↔ 𝑝 = (𝑥 substr ⟨0, 𝑁⟩)))
4039adantl 481 . . . . 5 ((((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) ∧ 𝑝 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑥𝐷) → (𝑝 = (𝐹𝑥) ↔ 𝑝 = (𝑥 substr ⟨0, 𝑁⟩)))
4140rexbidva 3031 . . . 4 (((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) ∧ 𝑝 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (∃𝑥𝐷 𝑝 = (𝐹𝑥) ↔ ∃𝑥𝐷 𝑝 = (𝑥 substr ⟨0, 𝑁⟩)))
4237, 41mpbird 246 . . 3 (((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) ∧ 𝑝 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ∃𝑥𝐷 𝑝 = (𝐹𝑥))
4342ralrimiva 2949 . 2 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ∀𝑝 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)∃𝑥𝐷 𝑝 = (𝐹𝑥))
44 dffo3 6282 . 2 (𝐹:𝐷onto→((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝐹:𝐷⟶((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∀𝑝 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)∃𝑥𝐷 𝑝 = (𝐹𝑥)))
453, 43, 44sylanbrc 695 1 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → 𝐹:𝐷onto→((𝑉 ClWWalksN 𝐸)‘𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  {cpr 4127  ⟨cop 4131   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕcn 10897  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150   WWalksN cwwlkn 26206   ClWWalksN cclwwlkn 26277 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 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-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-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  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-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-wwlk 26207  df-wwlkn 26208  df-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  clwwlkf1o  26326
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