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Theorem tgcgr4 25471
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 12588 . . . . 5 (0..^4) ⊆ ℕ0
6 nn0ssre 11334 . . . . 5 0 ⊆ ℝ
75, 6sstri 3645 . . . 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 13664 . . . . 5 (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑃)
14 wrdf 13342 . . . . 5 (⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑃 → ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^(#‘⟨“𝐴𝐵𝐶𝐷”⟩))⟶𝑃)
1513, 14syl 17 . . . 4 (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^(#‘⟨“𝐴𝐵𝐶𝐷”⟩))⟶𝑃)
16 s4len 13690 . . . . . 6 (#‘⟨“𝐴𝐵𝐶𝐷”⟩) = 4
1716oveq2i 6701 . . . . 5 (0..^(#‘⟨“𝐴𝐵𝐶𝐷”⟩)) = (0..^4)
1817feq2i 6075 . . . 4 (⟨“𝐴𝐵𝐶𝐷”⟩:(0..^(#‘⟨“𝐴𝐵𝐶𝐷”⟩))⟶𝑃 ↔ ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^4)⟶𝑃)
1915, 18sylib 208 . . 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 13664 . . . . 5 (𝜑 → ⟨“𝑊𝑋𝑌𝑍”⟩ ∈ Word 𝑃)
25 wrdf 13342 . . . . 5 (⟨“𝑊𝑋𝑌𝑍”⟩ ∈ Word 𝑃 → ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^(#‘⟨“𝑊𝑋𝑌𝑍”⟩))⟶𝑃)
2624, 25syl 17 . . . 4 (𝜑 → ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^(#‘⟨“𝑊𝑋𝑌𝑍”⟩))⟶𝑃)
27 s4len 13690 . . . . . 6 (#‘⟨“𝑊𝑋𝑌𝑍”⟩) = 4
2827oveq2i 6701 . . . . 5 (0..^(#‘⟨“𝑊𝑋𝑌𝑍”⟩)) = (0..^4)
2928feq2i 6075 . . . 4 (⟨“𝑊𝑋𝑌𝑍”⟩:(0..^(#‘⟨“𝑊𝑋𝑌𝑍”⟩))⟶𝑃 ↔ ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^4)⟶𝑃)
3026, 29sylib 208 . . 3 (𝜑 → ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^4)⟶𝑃)
311, 2, 3, 4, 8, 19, 30iscgrglt 25454 . 2 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩ ⟨“𝑊𝑋𝑌𝑍”⟩ ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)))))
32 fdm 6089 . . . . . . . 8 (⟨“𝐴𝐵𝐶𝐷”⟩:(0..^4)⟶𝑃 → dom ⟨“𝐴𝐵𝐶𝐷”⟩ = (0..^4))
3319, 32syl 17 . . . . . . 7 (𝜑 → dom ⟨“𝐴𝐵𝐶𝐷”⟩ = (0..^4))
34 3p1e4 11191 . . . . . . . . 9 (3 + 1) = 4
3534oveq2i 6701 . . . . . . . 8 (0..^(3 + 1)) = (0..^4)
36 3nn0 11348 . . . . . . . . . 10 3 ∈ ℕ0
37 nn0uz 11760 . . . . . . . . . 10 0 = (ℤ‘0)
3836, 37eleqtri 2728 . . . . . . . . 9 3 ∈ (ℤ‘0)
39 fzosplitsn 12616 . . . . . . . . 9 (3 ∈ (ℤ‘0) → (0..^(3 + 1)) = ((0..^3) ∪ {3}))
4038, 39ax-mp 5 . . . . . . . 8 (0..^(3 + 1)) = ((0..^3) ∪ {3})
4135, 40eqtr3i 2675 . . . . . . 7 (0..^4) = ((0..^3) ∪ {3})
4233, 41syl6eq 2701 . . . . . 6 (𝜑 → dom ⟨“𝐴𝐵𝐶𝐷”⟩ = ((0..^3) ∪ {3}))
4342raleqdv 3174 . . . . 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)))))
44 breq2 4689 . . . . . . . 8 (𝑗 = 3 → (𝑖 < 𝑗𝑖 < 3))
45 fveq2 6229 . . . . . . . . . 10 (𝑗 = 3 → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗) = (⟨“𝐴𝐵𝐶𝐷”⟩‘3))
4645oveq2d 6706 . . . . . . . . 9 (𝑗 = 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)))
47 fveq2 6229 . . . . . . . . . 10 (𝑗 = 3 → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗) = (⟨“𝑊𝑋𝑌𝑍”⟩‘3))
4847oveq2d 6706 . . . . . . . . 9 (𝑗 = 3 → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))
4946, 48eqeq12d 2666 . . . . . . . 8 (𝑗 = 3 → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))
5044, 49imbi12d 333 . . . . . . 7 (𝑗 = 3 → ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
5150ralunsn 4454 . . . . . 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))))))
5236, 51ax-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)))))
5343, 52syl6bb 276 . . . 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))))))
5453ralbidv 3015 . . 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))))))
5542raleqdv 3174 . . . 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))))))
56 fzo0ssnn0 12588 . . . . . . . . . . . . . . . 16 (0..^3) ⊆ ℕ0
5756, 6sstri 3645 . . . . . . . . . . . . . . 15 (0..^3) ⊆ ℝ
58 simpr 476 . . . . . . . . . . . . . . 15 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 ∈ (0..^3))
5957, 58sseldi 3634 . . . . . . . . . . . . . 14 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 ∈ ℝ)
60 simpl 472 . . . . . . . . . . . . . . 15 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑖 = 3)
616, 36sselii 3633 . . . . . . . . . . . . . . 15 3 ∈ ℝ
6260, 61syl6eqel 2738 . . . . . . . . . . . . . 14 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑖 ∈ ℝ)
63 elfzolt2 12518 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0..^3) → 𝑗 < 3)
6463adantl 481 . . . . . . . . . . . . . . 15 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 < 3)
6564, 60breqtrrd 4713 . . . . . . . . . . . . . 14 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 < 𝑖)
66 ltnsym 10173 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ℝ ∧ 𝑖 ∈ ℝ) → (𝑗 < 𝑖 → ¬ 𝑖 < 𝑗))
6766imp 444 . . . . . . . . . . . . . 14 (((𝑗 ∈ ℝ ∧ 𝑖 ∈ ℝ) ∧ 𝑗 < 𝑖) → ¬ 𝑖 < 𝑗)
6859, 62, 65, 67syl21anc 1365 . . . . . . . . . . . . 13 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → ¬ 𝑖 < 𝑗)
6968pm2.21d 118 . . . . . . . . . . . 12 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → (𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))))
70 tbtru 1534 . . . . . . . . . . . 12 ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⊤))
7169, 70sylib 208 . . . . . . . . . . 11 ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⊤))
7271ralbidva 3014 . . . . . . . . . 10 (𝑖 = 3 → (∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ∀𝑗 ∈ (0..^3)⊤))
73 3nn 11224 . . . . . . . . . . . . 13 3 ∈ ℕ
74 lbfzo0 12547 . . . . . . . . . . . . 13 (0 ∈ (0..^3) ↔ 3 ∈ ℕ)
7573, 74mpbir 221 . . . . . . . . . . . 12 0 ∈ (0..^3)
7675ne0ii 3956 . . . . . . . . . . 11 (0..^3) ≠ ∅
77 r19.3rzv 4097 . . . . . . . . . . 11 ((0..^3) ≠ ∅ → (⊤ ↔ ∀𝑗 ∈ (0..^3)⊤))
7876, 77ax-mp 5 . . . . . . . . . 10 (⊤ ↔ ∀𝑗 ∈ (0..^3)⊤)
7972, 78syl6bbr 278 . . . . . . . . 9 (𝑖 = 3 → (∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⊤))
80 breq1 4688 . . . . . . . . . . . 12 (𝑖 = 3 → (𝑖 < 3 ↔ 3 < 3))
8161ltnri 10184 . . . . . . . . . . . . 13 ¬ 3 < 3
8281bifal 1537 . . . . . . . . . . . 12 (3 < 3 ↔ ⊥)
8380, 82syl6bb 276 . . . . . . . . . . 11 (𝑖 = 3 → (𝑖 < 3 ↔ ⊥))
8483imbi1d 330 . . . . . . . . . 10 (𝑖 = 3 → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (⊥ → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
85 falim 1538 . . . . . . . . . . 11 (⊥ → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))
8685bitru 1536 . . . . . . . . . 10 ((⊥ → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ ⊤)
8784, 86syl6bb 276 . . . . . . . . 9 (𝑖 = 3 → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ ⊤))
8879, 87anbi12d 747 . . . . . . . 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)))) ↔ (⊤ ∧ ⊤)))
89 anidm 677 . . . . . . . 8 ((⊤ ∧ ⊤) ↔ ⊤)
9088, 89syl6bb 276 . . . . . . 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)))) ↔ ⊤))
9190ralunsn 4454 . . . . . 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)))) ∧ ⊤)))
9236, 91ax-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)))) ∧ ⊤))
93 ancom 465 . . . . 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))))))
94 truan 1541 . . . . 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)))))
9592, 93, 943bitri 286 . . . 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)))))
9655, 95syl6bb 276 . . 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))))))
9754, 96bitrd 268 . 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))))))
98 r19.26 3093 . . 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)))))
999, 10, 11s3cld 13663 . . . . . . . . 9 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
100 wrdf 13342 . . . . . . . . 9 (⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 → ⟨“𝐴𝐵𝐶”⟩:(0..^(#‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
10199, 100syl 17 . . . . . . . 8 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:(0..^(#‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
102 s3len 13685 . . . . . . . . . 10 (#‘⟨“𝐴𝐵𝐶”⟩) = 3
103102oveq2i 6701 . . . . . . . . 9 (0..^(#‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3)
104103feq2i 6075 . . . . . . . 8 (⟨“𝐴𝐵𝐶”⟩:(0..^(#‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃 ↔ ⟨“𝐴𝐵𝐶”⟩:(0..^3)⟶𝑃)
105101, 104sylib 208 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:(0..^3)⟶𝑃)
106 fdm 6089 . . . . . . 7 (⟨“𝐴𝐵𝐶”⟩:(0..^3)⟶𝑃 → dom ⟨“𝐴𝐵𝐶”⟩ = (0..^3))
107105, 106syl 17 . . . . . 6 (𝜑 → dom ⟨“𝐴𝐵𝐶”⟩ = (0..^3))
108 raleq 3168 . . . . . . 7 (dom ⟨“𝐴𝐵𝐶”⟩ = (0..^3) → (∀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)))))
109105, 106, 1083syl 18 . . . . . 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)))))
110107, 109raleqbidv 3182 . . . . 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)))))
11157a1i 11 . . . . . 6 (𝜑 → (0..^3) ⊆ ℝ)
11220, 21, 22s3cld 13663 . . . . . . . 8 (𝜑 → ⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃)
113 wrdf 13342 . . . . . . . 8 (⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃 → ⟨“𝑊𝑋𝑌”⟩:(0..^(#‘⟨“𝑊𝑋𝑌”⟩))⟶𝑃)
114112, 113syl 17 . . . . . . 7 (𝜑 → ⟨“𝑊𝑋𝑌”⟩:(0..^(#‘⟨“𝑊𝑋𝑌”⟩))⟶𝑃)
115 s3len 13685 . . . . . . . . 9 (#‘⟨“𝑊𝑋𝑌”⟩) = 3
116115oveq2i 6701 . . . . . . . 8 (0..^(#‘⟨“𝑊𝑋𝑌”⟩)) = (0..^3)
117116feq2i 6075 . . . . . . 7 (⟨“𝑊𝑋𝑌”⟩:(0..^(#‘⟨“𝑊𝑋𝑌”⟩))⟶𝑃 ↔ ⟨“𝑊𝑋𝑌”⟩:(0..^3)⟶𝑃)
118114, 117sylib 208 . . . . . 6 (𝜑 → ⟨“𝑊𝑋𝑌”⟩:(0..^3)⟶𝑃)
1191, 2, 3, 4, 111, 105, 118iscgrglt 25454 . . . . 5 (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝑊𝑋𝑌”⟩ ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗)))))
120 df-s4 13641 . . . . . . . . . . 11 ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
121120fveq1i 6230 . . . . . . . . . 10 (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑖)
1229adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐴𝑃)
12310adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐵𝑃)
12411adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐶𝑃)
125122, 123, 124s3cld 13663 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
12612adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐷𝑃)
127126s1cld 13419 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝐷”⟩ ∈ Word 𝑃)
128 simprl 809 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑖 ∈ (0..^3))
129128, 103syl6eleqr 2741 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑖 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩)))
130 ccatval1 13395 . . . . . . . . . . 11 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ ⟨“𝐷”⟩ ∈ Word 𝑃𝑖 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
131125, 127, 129, 130syl3anc 1366 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
132121, 131syl5eq 2697 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
133120fveq1i 6230 . . . . . . . . . 10 (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗) = ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑗)
134 simprr 811 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑗 ∈ (0..^3))
135134, 103syl6eleqr 2741 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑗 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩)))
136 ccatval1 13395 . . . . . . . . . . 11 ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ ⟨“𝐷”⟩ ∈ Word 𝑃𝑗 ∈ (0..^(#‘⟨“𝐴𝐵𝐶”⟩))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘𝑗))
137125, 127, 135, 136syl3anc 1366 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘𝑗))
138133, 137syl5eq 2697 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘𝑗))
139132, 138oveq12d 6708 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)))
140 df-s4 13641 . . . . . . . . . . 11 ⟨“𝑊𝑋𝑌𝑍”⟩ = (⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)
141140fveq1i 6230 . . . . . . . . . 10 (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑖)
14220adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑊𝑃)
14321adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑋𝑃)
14422adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑌𝑃)
145142, 143, 144s3cld 13663 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃)
14623adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑍𝑃)
147146s1cld 13419 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝑍”⟩ ∈ Word 𝑃)
148128, 116syl6eleqr 2741 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑖 ∈ (0..^(#‘⟨“𝑊𝑋𝑌”⟩)))
149 ccatval1 13395 . . . . . . . . . . 11 ((⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃 ∧ ⟨“𝑍”⟩ ∈ Word 𝑃𝑖 ∈ (0..^(#‘⟨“𝑊𝑋𝑌”⟩))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑖) = (⟨“𝑊𝑋𝑌”⟩‘𝑖))
150145, 147, 148, 149syl3anc 1366 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑖) = (⟨“𝑊𝑋𝑌”⟩‘𝑖))
151141, 150syl5eq 2697 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌”⟩‘𝑖))
152140fveq1i 6230 . . . . . . . . . 10 (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗) = ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑗)
153134, 116syl6eleqr 2741 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑗 ∈ (0..^(#‘⟨“𝑊𝑋𝑌”⟩)))
154 ccatval1 13395 . . . . . . . . . . 11 ((⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃 ∧ ⟨“𝑍”⟩ ∈ Word 𝑃𝑗 ∈ (0..^(#‘⟨“𝑊𝑋𝑌”⟩))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑗) = (⟨“𝑊𝑋𝑌”⟩‘𝑗))
155145, 147, 153, 154syl3anc 1366 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑗) = (⟨“𝑊𝑋𝑌”⟩‘𝑗))
156152, 155syl5eq 2697 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗) = (⟨“𝑊𝑋𝑌”⟩‘𝑗))
157151, 156oveq12d 6708 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗)))
158139, 157eqeq12d 2666 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) (⟨“𝑊𝑋𝑌”⟩‘𝑗))))
159158imbi2d 329 . . . . . 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)))))
1601592ralbidva 3017 . . . . 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)))))
161110, 119, 1603bitr4rd 301 . . . 4 (𝜑 → (∀𝑖 ∈ (0..^3)∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝑊𝑋𝑌”⟩))
162 fzo0to3tp 12594 . . . . . 6 (0..^3) = {0, 1, 2}
163 raleq 3168 . . . . . 6 ((0..^3) = {0, 1, 2} → (∀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)))))
164162, 163mp1i 13 . . . . 5 (𝜑 → (∀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)))))
165 3pos 11152 . . . . . . . . . 10 0 < 3
166 breq1 4688 . . . . . . . . . 10 (𝑖 = 0 → (𝑖 < 3 ↔ 0 < 3))
167165, 166mpbiri 248 . . . . . . . . 9 (𝑖 = 0 → 𝑖 < 3)
168167adantl 481 . . . . . . . 8 ((𝜑𝑖 = 0) → 𝑖 < 3)
169 biimt 349 . . . . . . . 8 (𝑖 < 3 → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
170168, 169syl 17 . . . . . . 7 ((𝜑𝑖 = 0) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
171 simpr 476 . . . . . . . . . . 11 ((𝜑𝑖 = 0) → 𝑖 = 0)
172171fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶𝐷”⟩‘0))
173 s4fv0 13686 . . . . . . . . . . . 12 (𝐴𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
1749, 173syl 17 . . . . . . . . . . 11 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
175174adantr 480 . . . . . . . . . 10 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
176172, 175eqtrd 2685 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = 𝐴)
177 s4fv3 13689 . . . . . . . . . . 11 (𝐷𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
17812, 177syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
179178adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
180176, 179oveq12d 6708 . . . . . . . 8 ((𝜑𝑖 = 0) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = (𝐴 𝐷))
181171fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑖 = 0) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌𝑍”⟩‘0))
182 s4fv0 13686 . . . . . . . . . . . 12 (𝑊𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘0) = 𝑊)
18320, 182syl 17 . . . . . . . . . . 11 (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘0) = 𝑊)
184183adantr 480 . . . . . . . . . 10 ((𝜑𝑖 = 0) → (⟨“𝑊𝑋𝑌𝑍”⟩‘0) = 𝑊)
185181, 184eqtrd 2685 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = 𝑊)
186 s4fv3 13689 . . . . . . . . . . 11 (𝑍𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
18723, 186syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
188187adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
189185, 188oveq12d 6708 . . . . . . . 8 ((𝜑𝑖 = 0) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) = (𝑊 𝑍))
190180, 189eqeq12d 2666 . . . . . . 7 ((𝜑𝑖 = 0) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝐴 𝐷) = (𝑊 𝑍)))
191170, 190bitr3d 270 . . . . . 6 ((𝜑𝑖 = 0) → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (𝐴 𝐷) = (𝑊 𝑍)))
192 1lt3 11234 . . . . . . . . . 10 1 < 3
193 breq1 4688 . . . . . . . . . 10 (𝑖 = 1 → (𝑖 < 3 ↔ 1 < 3))
194192, 193mpbiri 248 . . . . . . . . 9 (𝑖 = 1 → 𝑖 < 3)
195194adantl 481 . . . . . . . 8 ((𝜑𝑖 = 1) → 𝑖 < 3)
196195, 169syl 17 . . . . . . 7 ((𝜑𝑖 = 1) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
197 simpr 476 . . . . . . . . . . 11 ((𝜑𝑖 = 1) → 𝑖 = 1)
198197fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶𝐷”⟩‘1))
199 s4fv1 13687 . . . . . . . . . . . 12 (𝐵𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
20010, 199syl 17 . . . . . . . . . . 11 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
201200adantr 480 . . . . . . . . . 10 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
202198, 201eqtrd 2685 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = 𝐵)
203178adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
204202, 203oveq12d 6708 . . . . . . . 8 ((𝜑𝑖 = 1) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = (𝐵 𝐷))
205197fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑖 = 1) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌𝑍”⟩‘1))
206 s4fv1 13687 . . . . . . . . . . . 12 (𝑋𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘1) = 𝑋)
20721, 206syl 17 . . . . . . . . . . 11 (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘1) = 𝑋)
208207adantr 480 . . . . . . . . . 10 ((𝜑𝑖 = 1) → (⟨“𝑊𝑋𝑌𝑍”⟩‘1) = 𝑋)
209205, 208eqtrd 2685 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = 𝑋)
210187adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
211209, 210oveq12d 6708 . . . . . . . 8 ((𝜑𝑖 = 1) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) = (𝑋 𝑍))
212204, 211eqeq12d 2666 . . . . . . 7 ((𝜑𝑖 = 1) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝐵 𝐷) = (𝑋 𝑍)))
213196, 212bitr3d 270 . . . . . 6 ((𝜑𝑖 = 1) → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (𝐵 𝐷) = (𝑋 𝑍)))
214 2lt3 11233 . . . . . . . . . 10 2 < 3
215 breq1 4688 . . . . . . . . . 10 (𝑖 = 2 → (𝑖 < 3 ↔ 2 < 3))
216214, 215mpbiri 248 . . . . . . . . 9 (𝑖 = 2 → 𝑖 < 3)
217216adantl 481 . . . . . . . 8 ((𝜑𝑖 = 2) → 𝑖 < 3)
218217, 169syl 17 . . . . . . 7 ((𝜑𝑖 = 2) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
219 simpr 476 . . . . . . . . . . 11 ((𝜑𝑖 = 2) → 𝑖 = 2)
220219fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶𝐷”⟩‘2))
221 s4fv2 13688 . . . . . . . . . . . 12 (𝐶𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
22211, 221syl 17 . . . . . . . . . . 11 (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
223222adantr 480 . . . . . . . . . 10 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
224220, 223eqtrd 2685 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = 𝐶)
225178adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
226224, 225oveq12d 6708 . . . . . . . 8 ((𝜑𝑖 = 2) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = (𝐶 𝐷))
227219fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑖 = 2) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌𝑍”⟩‘2))
228 s4fv2 13688 . . . . . . . . . . . 12 (𝑌𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘2) = 𝑌)
22922, 228syl 17 . . . . . . . . . . 11 (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘2) = 𝑌)
230229adantr 480 . . . . . . . . . 10 ((𝜑𝑖 = 2) → (⟨“𝑊𝑋𝑌𝑍”⟩‘2) = 𝑌)
231227, 230eqtrd 2685 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = 𝑌)
232187adantr 480 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
233231, 232oveq12d 6708 . . . . . . . 8 ((𝜑𝑖 = 2) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) = (𝑌 𝑍))
234226, 233eqeq12d 2666 . . . . . . 7 ((𝜑𝑖 = 2) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝐶 𝐷) = (𝑌 𝑍)))
235218, 234bitr3d 270 . . . . . 6 ((𝜑𝑖 = 2) → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (𝐶 𝐷) = (𝑌 𝑍)))
236 0red 10079 . . . . . 6 (𝜑 → 0 ∈ ℝ)
237 1red 10093 . . . . . 6 (𝜑 → 1 ∈ ℝ)
238 2re 11128 . . . . . . 7 2 ∈ ℝ
239238a1i 11 . . . . . 6 (𝜑 → 2 ∈ ℝ)
240191, 213, 235, 236, 237, 239raltpd 4346 . . . . 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))))
241164, 240bitrd 268 . . . 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))))
242161, 241anbi12d 747 . . 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)))))
24398, 242syl5bb 272 . 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)))))
24431, 97, 2433bitrd 294 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:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wtru 1524  wfal 1528  wcel 2030  wne 2823  wral 2941  cun 3605  wss 3607  c0 3948  {csn 4210  {ctp 4214   class class class wbr 4685  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cn 11058  2c2 11108  3c3 11109  4c4 11110  0cn0 11330  cuz 11725  ..^cfzo 12504  #chash 13157  Word cword 13323   ++ cconcat 13325  ⟨“cs1 13326  ⟨“cs3 13633  ⟨“cs4 13634  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  cgrGccgrg 25450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-s4 13641  df-trkgc 25392  df-trkgcb 25394  df-trkg 25397  df-cgrg 25451
This theorem is referenced by:  cgrg3col4  25779
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