MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgcgr4 Structured version   Visualization version   GIF version

Theorem tgcgr4 26890
Description: Two quadrilaterals to be congruent to each other if one triangle formed by their vertices is, and the additional points are equidistant too. (Contributed by Thierry Arnoux, 8-Oct-2020.)
Hypotheses
Ref Expression
tgcgrxfr.p 𝑃 = (Base‘𝐺)
tgcgrxfr.m = (dist‘𝐺)
tgcgrxfr.i 𝐼 = (Itv‘𝐺)
tgcgrxfr.r = (cgrG‘𝐺)
tgcgrxfr.g (𝜑𝐺 ∈ TarskiG)
tgcgr4.a (𝜑𝐴𝑃)
tgcgr4.b (𝜑𝐵𝑃)
tgcgr4.c (𝜑𝐶𝑃)
tgcgr4.d (𝜑𝐷𝑃)
tgcgr4.w (𝜑𝑊𝑃)
tgcgr4.x (𝜑𝑋𝑃)
tgcgr4.y (𝜑𝑌𝑃)
tgcgr4.z (𝜑𝑍𝑃)
Assertion
Ref Expression
tgcgr4 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩ ⟨“𝑊𝑋𝑌𝑍”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩ ⟨“𝑊𝑋𝑌”⟩ ∧ ((𝐴 𝐷) = (𝑊 𝑍) ∧ (𝐵 𝐷) = (𝑋 𝑍) ∧ (𝐶 𝐷) = (𝑌 𝑍)))))

Proof of Theorem tgcgr4
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgcgrxfr.p . . 3 𝑃 = (Base‘𝐺)
2 tgcgrxfr.m . . 3 = (dist‘𝐺)
3 tgcgrxfr.r . . 3 = (cgrG‘𝐺)
4 tgcgrxfr.g . . 3 (𝜑𝐺 ∈ TarskiG)
5 fzo0ssnn0 13466 . . . . 5 (0..^4) ⊆ ℕ0
6 nn0ssre 12237 . . . . 5 0 ⊆ ℝ
75, 6sstri 3935 . . . 4 (0..^4) ⊆ ℝ
87a1i 11 . . 3 (𝜑 → (0..^4) ⊆ ℝ)
9 tgcgr4.a . . . . . 6 (𝜑𝐴𝑃)
10 tgcgr4.b . . . . . 6 (𝜑𝐵𝑃)
11 tgcgr4.c . . . . . 6 (𝜑𝐶𝑃)
12 tgcgr4.d . . . . . 6 (𝜑𝐷𝑃)
139, 10, 11, 12s4cld 14584 . . . . 5 (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑃)
14 wrdf 14220 . . . . 5 (⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑃 → ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶𝐷”⟩))⟶𝑃)
1513, 14syl 17 . . . 4 (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶𝐷”⟩))⟶𝑃)
16 s4len 14610 . . . . . 6 (♯‘⟨“𝐴𝐵𝐶𝐷”⟩) = 4
1716oveq2i 7282 . . . . 5 (0..^(♯‘⟨“𝐴𝐵𝐶𝐷”⟩)) = (0..^4)
1817feq2i 6590 . . . 4 (⟨“𝐴𝐵𝐶𝐷”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶𝐷”⟩))⟶𝑃 ↔ ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^4)⟶𝑃)
1915, 18sylib 217 . . 3 (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^4)⟶𝑃)
20 tgcgr4.w . . . . . 6 (𝜑𝑊𝑃)
21 tgcgr4.x . . . . . 6 (𝜑𝑋𝑃)
22 tgcgr4.y . . . . . 6 (𝜑𝑌𝑃)
23 tgcgr4.z . . . . . 6 (𝜑𝑍𝑃)
2420, 21, 22, 23s4cld 14584 . . . . 5 (𝜑 → ⟨“𝑊𝑋𝑌𝑍”⟩ ∈ Word 𝑃)
25 wrdf 14220 . . . . 5 (⟨“𝑊𝑋𝑌𝑍”⟩ ∈ Word 𝑃 → ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌𝑍”⟩))⟶𝑃)
2624, 25syl 17 . . . 4 (𝜑 → ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌𝑍”⟩))⟶𝑃)
27 s4len 14610 . . . . . 6 (♯‘⟨“𝑊𝑋𝑌𝑍”⟩) = 4
2827oveq2i 7282 . . . . 5 (0..^(♯‘⟨“𝑊𝑋𝑌𝑍”⟩)) = (0..^4)
2928feq2i 6590 . . . 4 (⟨“𝑊𝑋𝑌𝑍”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌𝑍”⟩))⟶𝑃 ↔ ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^4)⟶𝑃)
3026, 29sylib 217 . . 3 (𝜑 → ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^4)⟶𝑃)
311, 2, 3, 4, 8, 19, 30iscgrglt 26873 . 2 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩ ⟨“𝑊𝑋𝑌𝑍”⟩ ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)))))
3219fdmd 6609 . . . . . . 7 (𝜑 → dom ⟨“𝐴𝐵𝐶𝐷”⟩ = (0..^4))
33 3p1e4 12118 . . . . . . . . 9 (3 + 1) = 4
3433oveq2i 7282 . . . . . . . 8 (0..^(3 + 1)) = (0..^4)
35 3nn0 12251 . . . . . . . . . 10 3 ∈ ℕ0
36 nn0uz 12619 . . . . . . . . . 10 0 = (ℤ‘0)
3735, 36eleqtri 2839 . . . . . . . . 9 3 ∈ (ℤ‘0)
38 fzosplitsn 13493 . . . . . . . . 9 (3 ∈ (ℤ‘0) → (0..^(3 + 1)) = ((0..^3) ∪ {3}))
3937, 38ax-mp 5 . . . . . . . 8 (0..^(3 + 1)) = ((0..^3) ∪ {3})
4034, 39eqtr3i 2770 . . . . . . 7 (0..^4) = ((0..^3) ∪ {3})
4132, 40eqtrdi 2796 . . . . . 6 (𝜑 → dom ⟨“𝐴𝐵𝐶𝐷”⟩ = ((0..^3) ∪ {3}))
4241raleqdv 3347 . . . . 5 (𝜑 → (∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ∀𝑗 ∈ ((0..^3) ∪ {3})(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)))))
43 breq2 5083 . . . . . . . 8 (𝑗 = 3 → (𝑖 < 𝑗𝑖 < 3))
44 fveq2 6771 . . . . . . . . . 10 (𝑗 = 3 → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗) = (⟨“𝐴𝐵𝐶𝐷”⟩‘3))
4544oveq2d 7287 . . . . . . . . 9 (𝑗 = 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)))
46 fveq2 6771 . . . . . . . . . 10 (𝑗 = 3 → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗) = (⟨“𝑊𝑋𝑌𝑍”⟩‘3))
4746oveq2d 7287 . . . . . . . . 9 (𝑗 = 3 → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))
4845, 47eqeq12d 2756 . . . . . . . 8 (𝑗 = 3 → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))
4943, 48imbi12d 345 . . . . . . 7 (𝑗 = 3 → ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
5049ralunsn 4831 . . . . . 6 (3 ∈ ℕ0 → (∀𝑗 ∈ ((0..^3) ∪ {3})(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ (∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))))
5135, 50ax-mp 5 . . . . 5 (∀𝑗 ∈ ((0..^3) ∪ {3})(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ (∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
5242, 51bitrdi 287 . . . 4 (𝜑 → (∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ (∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))))
5352ralbidv 3123 . . 3 (𝜑 → (∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))))
5441raleqdv 3347 . . . 4 (𝜑 → (∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ↔ ∀𝑖 ∈ ((0..^3) ∪ {3})(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))))
55 fzo0ssnn0 13466 . . . . . . . . . . . . . . . 16 (0..^3) ⊆ ℕ0
5655, 6sstri 3935 . . . . . . . . . . . . . . 15 (0..^3) ⊆ ℝ
57 simpr 485 . . . . . . . . . . . . . . 15 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 ∈ (0..^3))
5856, 57sselid 3924 . . . . . . . . . . . . . 14 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 ∈ ℝ)
59 simpl 483 . . . . . . . . . . . . . . 15 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑖 = 3)
60 3re 12053 . . . . . . . . . . . . . . 15 3 ∈ ℝ
6159, 60eqeltrdi 2849 . . . . . . . . . . . . . 14 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑖 ∈ ℝ)
62 elfzolt2 13395 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0..^3) → 𝑗 < 3)
6362adantl 482 . . . . . . . . . . . . . . 15 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 < 3)
6463, 59breqtrrd 5107 . . . . . . . . . . . . . 14 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 < 𝑖)
6558, 61, 64ltnsymd 11124 . . . . . . . . . . . . 13 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → ¬ 𝑖 < 𝑗)
6665pm2.21d 121 . . . . . . . . . . . 12 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → (𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))))
67 tbtru 1550 . . . . . . . . . . . 12 ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⊤))
6866, 67sylib 217 . . . . . . . . . . 11 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⊤))
6968ralbidva 3122 . . . . . . . . . 10 (𝑖 = 3 → (∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ∀𝑗 ∈ (0..^3)⊤))
70 3nn 12052 . . . . . . . . . . . . 13 3 ∈ ℕ
71 lbfzo0 13425 . . . . . . . . . . . . 13 (0 ∈ (0..^3) ↔ 3 ∈ ℕ)
7270, 71mpbir 230 . . . . . . . . . . . 12 0 ∈ (0..^3)
7372ne0ii 4277 . . . . . . . . . . 11 (0..^3) ≠ ∅
74 r19.3rzv 4435 . . . . . . . . . . 11 ((0..^3) ≠ ∅ → (⊤ ↔ ∀𝑗 ∈ (0..^3)⊤))
7573, 74ax-mp 5 . . . . . . . . . 10 (⊤ ↔ ∀𝑗 ∈ (0..^3)⊤)
7669, 75bitr4di 289 . . . . . . . . 9 (𝑖 = 3 → (∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⊤))
77 breq1 5082 . . . . . . . . . . . 12 (𝑖 = 3 → (𝑖 < 3 ↔ 3 < 3))
7860ltnri 11084 . . . . . . . . . . . . 13 ¬ 3 < 3
7978bifal 1558 . . . . . . . . . . . 12 (3 < 3 ↔ ⊥)
8077, 79bitrdi 287 . . . . . . . . . . 11 (𝑖 = 3 → (𝑖 < 3 ↔ ⊥))
8180imbi1d 342 . . . . . . . . . 10 (𝑖 = 3 → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (⊥ → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
82 falim 1559 . . . . . . . . . . 11 (⊥ → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))
8382bitru 1551 . . . . . . . . . 10 ((⊥ → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ ⊤)
8481, 83bitrdi 287 . . . . . . . . 9 (𝑖 = 3 → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ ⊤))
8576, 84anbi12d 631 . . . . . . . 8 (𝑖 = 3 → ((∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ↔ (⊤ ∧ ⊤)))
86 anidm 565 . . . . . . . 8 ((⊤ ∧ ⊤) ↔ ⊤)
8785, 86bitrdi 287 . . . . . . 7 (𝑖 = 3 → ((∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ↔ ⊤))
8887ralunsn 4831 . . . . . 6 (3 ∈ ℕ0 → (∀𝑖 ∈ ((0..^3) ∪ {3})(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ↔ (∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ∧ ⊤)))
8935, 88ax-mp 5 . . . . 5 (∀𝑖 ∈ ((0..^3) ∪ {3})(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ↔ (∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ∧ ⊤))
90 ancom 461 . . . . 5 ((∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ∧ ⊤) ↔ (⊤ ∧ ∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))))
91 truan 1553 . . . . 5 ((⊤ ∧ ∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))) ↔ ∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
9289, 90, 913bitri 297 . . . 4 (∀𝑖 ∈ ((0..^3) ∪ {3})(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ↔ ∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
9354, 92bitrdi 287 . . 3 (𝜑 → (∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ↔ ∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))))
9453, 93bitrd 278 . 2 (𝜑 → (∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))))
95 r19.26 3097 . . 3 (∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ↔ (∀𝑖 ∈ (0..^3)∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ ∀𝑖 ∈ (0..^3)(𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
969, 10, 11s3cld 14583 . . . . . . . . 9 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
97 wrdf 14220 . . . . . . . . 9 (⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 → ⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
9896, 97syl 17 . . . . . . . 8 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
99 s3len 14605 . . . . . . . . . 10 (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
10099oveq2i 7282 . . . . . . . . 9 (0..^(♯‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3)
101100feq2i 6590 . . . . . . . 8 (⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃 ↔ ⟨“𝐴𝐵𝐶”⟩:(0..^3)⟶𝑃)
10298, 101sylib 217 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:(0..^3)⟶𝑃)
103102fdmd 6609 . . . . . 6 (𝜑 → dom ⟨“𝐴𝐵𝐶”⟩ = (0..^3))
104103raleqdv 3347 . . . . . 6 (𝜑 → (∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗))) ↔ ∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗)))))
105103, 104raleqbidv 3335 . . . . 5 (𝜑 → (∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗))) ↔ ∀𝑖 ∈ (0..^3)∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗)))))
10656a1i 11 . . . . . 6 (𝜑 → (0..^3) ⊆ ℝ)
10720, 21, 22s3cld 14583 . . . . . . . 8 (𝜑 → ⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃)
108 wrdf 14220 . . . . . . . 8 (⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃 → ⟨“𝑊𝑋𝑌”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌”⟩))⟶𝑃)
109107, 108syl 17 . . . . . . 7 (𝜑 → ⟨“𝑊𝑋𝑌”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌”⟩))⟶𝑃)
110 s3len 14605 . . . . . . . . 9 (♯‘⟨“𝑊𝑋𝑌”⟩) = 3
111110oveq2i 7282 . . . . . . . 8 (0..^(♯‘⟨“𝑊𝑋𝑌”⟩)) = (0..^3)
112111feq2i 6590 . . . . . . 7 (⟨“𝑊𝑋𝑌”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌”⟩))⟶𝑃 ↔ ⟨“𝑊𝑋𝑌”⟩:(0..^3)⟶𝑃)
113109, 112sylib 217 . . . . . 6 (𝜑 → ⟨“𝑊𝑋𝑌”⟩:(0..^3)⟶𝑃)
1141, 2, 3, 4, 106, 102, 113iscgrglt 26873 . . . . 5 (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝑊𝑋𝑌”⟩ ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗)))))
115 df-s4 14561 . . . . . . . . . . 11 ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
116115fveq1i 6772 . . . . . . . . . 10 (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑖)
1179adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐴𝑃)
11810adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐵𝑃)
11911adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐶𝑃)
120117, 118, 119s3cld 14583 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
12112adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐷𝑃)
122121s1cld 14306 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝐷”⟩ ∈ Word 𝑃)
123 simprl 768 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑖 ∈ (0..^3))
124123, 100eleqtrrdi 2852 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑖 ∈ (0..^(♯‘⟨“𝐴𝐵𝐶”⟩)))
125 ccatval1 14279 . . . . . . . . . . 11 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ ⟨“𝐷”⟩ ∈ Word 𝑃𝑖 ∈ (0..^(♯‘⟨“𝐴𝐵𝐶”⟩))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
126120, 122, 124, 125syl3anc 1370 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
127116, 126eqtrid 2792 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
128115fveq1i 6772 . . . . . . . . . 10 (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗) = ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑗)
129 simprr 770 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑗 ∈ (0..^3))
130129, 100eleqtrrdi 2852 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑗 ∈ (0..^(♯‘⟨“𝐴𝐵𝐶”⟩)))
131 ccatval1 14279 . . . . . . . . . . 11 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ ⟨“𝐷”⟩ ∈ Word 𝑃𝑗 ∈ (0..^(♯‘⟨“𝐴𝐵𝐶”⟩))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘𝑗))
132120, 122, 130, 131syl3anc 1370 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘𝑗))
133128, 132eqtrid 2792 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘𝑗))
134127, 133oveq12d 7289 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)))
135 df-s4 14561 . . . . . . . . . . 11 ⟨“𝑊𝑋𝑌𝑍”⟩ = (⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)
136135fveq1i 6772 . . . . . . . . . 10 (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑖)
13720adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑊𝑃)
13821adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑋𝑃)
13922adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑌𝑃)
140137, 138, 139s3cld 14583 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃)
14123adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑍𝑃)
142141s1cld 14306 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝑍”⟩ ∈ Word 𝑃)
143123, 111eleqtrrdi 2852 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑖 ∈ (0..^(♯‘⟨“𝑊𝑋𝑌”⟩)))
144 ccatval1 14279 . . . . . . . . . . 11 ((⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃 ∧ ⟨“𝑍”⟩ ∈ Word 𝑃𝑖 ∈ (0..^(♯‘⟨“𝑊𝑋𝑌”⟩))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑖) = (⟨“𝑊𝑋𝑌”⟩‘𝑖))
145140, 142, 143, 144syl3anc 1370 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑖) = (⟨“𝑊𝑋𝑌”⟩‘𝑖))
146136, 145eqtrid 2792 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌”⟩‘𝑖))
147135fveq1i 6772 . . . . . . . . . 10 (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗) = ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑗)
148129, 111eleqtrrdi 2852 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑗 ∈ (0..^(♯‘⟨“𝑊𝑋𝑌”⟩)))
149 ccatval1 14279 . . . . . . . . . . 11 ((⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃 ∧ ⟨“𝑍”⟩ ∈ Word 𝑃𝑗 ∈ (0..^(♯‘⟨“𝑊𝑋𝑌”⟩))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑗) = (⟨“𝑊𝑋𝑌”⟩‘𝑗))
150140, 142, 148, 149syl3anc 1370 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑗) = (⟨“𝑊𝑋𝑌”⟩‘𝑗))
151147, 150eqtrid 2792 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗) = (⟨“𝑊𝑋𝑌”⟩‘𝑗))
152146, 151oveq12d 7289 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗)))
153134, 152eqeq12d 2756 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗))))
154153imbi2d 341 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ (𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗)))))
1551542ralbidva 3124 . . . . 5 (𝜑 → (∀𝑖 ∈ (0..^3)∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ∀𝑖 ∈ (0..^3)∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗)))))
156105, 114, 1553bitr4rd 312 . . . 4 (𝜑 → (∀𝑖 ∈ (0..^3)∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝑊𝑋𝑌”⟩))
157 fzo0to3tp 13471 . . . . . 6 (0..^3) = {0, 1, 2}
158157raleqi 3345 . . . . 5 (∀𝑖 ∈ (0..^3)(𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ ∀𝑖 ∈ {0, 1, 2} (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))
159 3pos 12078 . . . . . . . . . 10 0 < 3
160 breq1 5082 . . . . . . . . . 10 (𝑖 = 0 → (𝑖 < 3 ↔ 0 < 3))
161159, 160mpbiri 257 . . . . . . . . 9 (𝑖 = 0 → 𝑖 < 3)
162161adantl 482 . . . . . . . 8 ((𝜑𝑖 = 0) → 𝑖 < 3)
163 biimt 361 . . . . . . . 8 (𝑖 < 3 → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
164162, 163syl 17 . . . . . . 7 ((𝜑𝑖 = 0) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
165 fveq2 6771 . . . . . . . . . 10 (𝑖 = 0 → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶𝐷”⟩‘0))
166 s4fv0 14606 . . . . . . . . . . 11 (𝐴𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
1679, 166syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
168165, 167sylan9eqr 2802 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = 𝐴)
169 s4fv3 14609 . . . . . . . . . . 11 (𝐷𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
17012, 169syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
171170adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
172168, 171oveq12d 7289 . . . . . . . 8 ((𝜑𝑖 = 0) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = (𝐴 𝐷))
173 fveq2 6771 . . . . . . . . . 10 (𝑖 = 0 → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌𝑍”⟩‘0))
174 s4fv0 14606 . . . . . . . . . . 11 (𝑊𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘0) = 𝑊)
17520, 174syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘0) = 𝑊)
176173, 175sylan9eqr 2802 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = 𝑊)
177 s4fv3 14609 . . . . . . . . . . 11 (𝑍𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
17823, 177syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
179178adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
180176, 179oveq12d 7289 . . . . . . . 8 ((𝜑𝑖 = 0) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) = (𝑊 𝑍))
181172, 180eqeq12d 2756 . . . . . . 7 ((𝜑𝑖 = 0) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝐴 𝐷) = (𝑊 𝑍)))
182164, 181bitr3d 280 . . . . . 6 ((𝜑𝑖 = 0) → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (𝐴 𝐷) = (𝑊 𝑍)))
183 1lt3 12146 . . . . . . . . . 10 1 < 3
184 breq1 5082 . . . . . . . . . 10 (𝑖 = 1 → (𝑖 < 3 ↔ 1 < 3))
185183, 184mpbiri 257 . . . . . . . . 9 (𝑖 = 1 → 𝑖 < 3)
186185adantl 482 . . . . . . . 8 ((𝜑𝑖 = 1) → 𝑖 < 3)
187186, 163syl 17 . . . . . . 7 ((𝜑𝑖 = 1) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
188 fveq2 6771 . . . . . . . . . 10 (𝑖 = 1 → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶𝐷”⟩‘1))
189 s4fv1 14607 . . . . . . . . . . 11 (𝐵𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
19010, 189syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
191188, 190sylan9eqr 2802 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = 𝐵)
192170adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
193191, 192oveq12d 7289 . . . . . . . 8 ((𝜑𝑖 = 1) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = (𝐵 𝐷))
194 fveq2 6771 . . . . . . . . . 10 (𝑖 = 1 → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌𝑍”⟩‘1))
195 s4fv1 14607 . . . . . . . . . . 11 (𝑋𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘1) = 𝑋)
19621, 195syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘1) = 𝑋)
197194, 196sylan9eqr 2802 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = 𝑋)
198178adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
199197, 198oveq12d 7289 . . . . . . . 8 ((𝜑𝑖 = 1) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) = (𝑋 𝑍))
200193, 199eqeq12d 2756 . . . . . . 7 ((𝜑𝑖 = 1) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝐵 𝐷) = (𝑋 𝑍)))
201187, 200bitr3d 280 . . . . . 6 ((𝜑𝑖 = 1) → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (𝐵 𝐷) = (𝑋 𝑍)))
202 2lt3 12145 . . . . . . . . . 10 2 < 3
203 breq1 5082 . . . . . . . . . 10 (𝑖 = 2 → (𝑖 < 3 ↔ 2 < 3))
204202, 203mpbiri 257 . . . . . . . . 9 (𝑖 = 2 → 𝑖 < 3)
205204adantl 482 . . . . . . . 8 ((𝜑𝑖 = 2) → 𝑖 < 3)
206205, 163syl 17 . . . . . . 7 ((𝜑𝑖 = 2) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
207 fveq2 6771 . . . . . . . . . 10 (𝑖 = 2 → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶𝐷”⟩‘2))
208 s4fv2 14608 . . . . . . . . . . 11 (𝐶𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
20911, 208syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
210207, 209sylan9eqr 2802 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = 𝐶)
211170adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
212210, 211oveq12d 7289 . . . . . . . 8 ((𝜑𝑖 = 2) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = (𝐶 𝐷))
213 fveq2 6771 . . . . . . . . . 10 (𝑖 = 2 → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌𝑍”⟩‘2))
214 s4fv2 14608 . . . . . . . . . . 11 (𝑌𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘2) = 𝑌)
21522, 214syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘2) = 𝑌)
216213, 215sylan9eqr 2802 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = 𝑌)
217178adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
218216, 217oveq12d 7289 . . . . . . . 8 ((𝜑𝑖 = 2) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) = (𝑌 𝑍))
219212, 218eqeq12d 2756 . . . . . . 7 ((𝜑𝑖 = 2) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝐶 𝐷) = (𝑌 𝑍)))
220206, 219bitr3d 280 . . . . . 6 ((𝜑𝑖 = 2) → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (𝐶 𝐷) = (𝑌 𝑍)))
221 0red 10979 . . . . . 6 (𝜑 → 0 ∈ ℝ)
222 1red 10977 . . . . . 6 (𝜑 → 1 ∈ ℝ)
223 2re 12047 . . . . . . 7 2 ∈ ℝ
224223a1i 11 . . . . . 6 (𝜑 → 2 ∈ ℝ)
225182, 201, 220, 221, 222, 224raltpd 4723 . . . . 5 (𝜑 → (∀𝑖 ∈ {0, 1, 2} (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ ((𝐴 𝐷) = (𝑊 𝑍) ∧ (𝐵 𝐷) = (𝑋 𝑍) ∧ (𝐶 𝐷) = (𝑌 𝑍))))
226158, 225syl5bb 283 . . . 4 (𝜑 → (∀𝑖 ∈ (0..^3)(𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ ((𝐴 𝐷) = (𝑊 𝑍) ∧ (𝐵 𝐷) = (𝑋 𝑍) ∧ (𝐶 𝐷) = (𝑌 𝑍))))
227156, 226anbi12d 631 . . 3 (𝜑 → ((∀𝑖 ∈ (0..^3)∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ ∀𝑖 ∈ (0..^3)(𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ↔ (⟨“𝐴𝐵𝐶”⟩ ⟨“𝑊𝑋𝑌”⟩ ∧ ((𝐴 𝐷) = (𝑊 𝑍) ∧ (𝐵 𝐷) = (𝑋 𝑍) ∧ (𝐶 𝐷) = (𝑌 𝑍)))))
22895, 227syl5bb 283 . 2 (𝜑 → (∀𝑖 ∈ (0..^3)(∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ∧ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))) ↔ (⟨“𝐴𝐵𝐶”⟩ ⟨“𝑊𝑋𝑌”⟩ ∧ ((𝐴 𝐷) = (𝑊 𝑍) ∧ (𝐵 𝐷) = (𝑋 𝑍) ∧ (𝐶 𝐷) = (𝑌 𝑍)))))
22931, 94, 2283bitrd 305 1 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩ ⟨“𝑊𝑋𝑌𝑍”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩ ⟨“𝑊𝑋𝑌”⟩ ∧ ((𝐴 𝐷) = (𝑊 𝑍) ∧ (𝐵 𝐷) = (𝑋 𝑍) ∧ (𝐶 𝐷) = (𝑌 𝑍)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1542  wtru 1543  wfal 1554  wcel 2110  wne 2945  wral 3066  cun 3890  wss 3892  c0 4262  {csn 4567  {ctp 4571   class class class wbr 5079  dom cdm 5590  wf 6428  cfv 6432  (class class class)co 7271  cr 10871  0cc0 10872  1c1 10873   + caddc 10875   < clt 11010  cn 11973  2c2 12028  3c3 12029  4c4 12030  0cn0 12233  cuz 12581  ..^cfzo 13381  chash 14042  Word cword 14215   ++ cconcat 14271  ⟨“cs1 14298  ⟨“cs3 14553  ⟨“cs4 14554  Basecbs 16910  distcds 16969  TarskiGcstrkg 26786  Itvcitv 26792  cgrGccgrg 26869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-er 8481  df-pm 8601  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-card 9698  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-n0 12234  df-z 12320  df-uz 12582  df-fz 13239  df-fzo 13382  df-hash 14043  df-word 14216  df-concat 14272  df-s1 14299  df-s2 14559  df-s3 14560  df-s4 14561  df-trkgc 26807  df-trkgcb 26809  df-trkg 26812  df-cgrg 26870
This theorem is referenced by:  cgrg3col4  27212
  Copyright terms: Public domain W3C validator