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Theorem tgcgr4 26034

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgcgr4.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgcgr4.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgcgr4.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgcgr4.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgcgr4.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
tgcgr4.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tgcgr4.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgcgr4.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)

**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.

Step	Hyp	Ref	Expression
1		tgcgrxfr.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		tgcgrxfr.m	. . 3 ⊢ − = (dist‘𝐺)
3		tgcgrxfr.r	. . 3 ⊢ ∼ = (cgrG‘𝐺)
4		tgcgrxfr.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		fzo0ssnn0 12939	. . . . 5 ⊢ (0..^4) ⊆ ℕ₀
6		nn0ssre 11717	. . . . 5 ⊢ ℕ₀ ⊆ ℝ
7	5, 6	sstri 3869	. . . 4 ⊢ (0..^4) ⊆ ℝ
8	7	a1i 11	. . 3 ⊢ (𝜑 → (0..^4) ⊆ ℝ)
9		tgcgr4.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
10		tgcgr4.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11		tgcgr4.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
12		tgcgr4.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13	9, 10, 11, 12	s4cld 14103	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑃)
14		wrdf 13683	. . . . 5 ⊢ (⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑃 → ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶𝐷”⟩))⟶𝑃)
15	13, 14	syl 17	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶𝐷”⟩))⟶𝑃)
16		s4len 14129	. . . . . 6 ⊢ (♯‘⟨“𝐴𝐵𝐶𝐷”⟩) = 4
17	16	oveq2i 6993	. . . . 5 ⊢ (0..^(♯‘⟨“𝐴𝐵𝐶𝐷”⟩)) = (0..^4)
18	17	feq2i 6341	. . . 4 ⊢ (⟨“𝐴𝐵𝐶𝐷”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶𝐷”⟩))⟶𝑃 ↔ ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^4)⟶𝑃)
19	15, 18	sylib 210	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩:(0..^4)⟶𝑃)
20		tgcgr4.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
21		tgcgr4.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
22		tgcgr4.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
23		tgcgr4.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
24	20, 21, 22, 23	s4cld 14103	. . . . 5 ⊢ (𝜑 → ⟨“𝑊𝑋𝑌𝑍”⟩ ∈ Word 𝑃)
25		wrdf 13683	. . . . 5 ⊢ (⟨“𝑊𝑋𝑌𝑍”⟩ ∈ Word 𝑃 → ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌𝑍”⟩))⟶𝑃)
26	24, 25	syl 17	. . . 4 ⊢ (𝜑 → ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌𝑍”⟩))⟶𝑃)
27		s4len 14129	. . . . . 6 ⊢ (♯‘⟨“𝑊𝑋𝑌𝑍”⟩) = 4
28	27	oveq2i 6993	. . . . 5 ⊢ (0..^(♯‘⟨“𝑊𝑋𝑌𝑍”⟩)) = (0..^4)
29	28	feq2i 6341	. . . 4 ⊢ (⟨“𝑊𝑋𝑌𝑍”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌𝑍”⟩))⟶𝑃 ↔ ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^4)⟶𝑃)
30	26, 29	sylib 210	. . 3 ⊢ (𝜑 → ⟨“𝑊𝑋𝑌𝑍”⟩:(0..^4)⟶𝑃)
31	1, 2, 3, 4, 8, 19, 30	iscgrglt 26017	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩ ∼ ⟨“𝑊𝑋𝑌𝑍”⟩ ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶𝐷”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)))))
32	19	fdmd 6358	. . . . . . 7 ⊢ (𝜑 → dom ⟨“𝐴𝐵𝐶𝐷”⟩ = (0..^4))
33		3p1e4 11598	. . . . . . . . 9 ⊢ (3 + 1) = 4
34	33	oveq2i 6993	. . . . . . . 8 ⊢ (0..^(3 + 1)) = (0..^4)
35		3nn0 11733	. . . . . . . . . 10 ⊢ 3 ∈ ℕ₀
36		nn0uz 12100	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
37	35, 36	eleqtri 2866	. . . . . . . . 9 ⊢ 3 ∈ (ℤ_≥‘0)
38		fzosplitsn 12966	. . . . . . . . 9 ⊢ (3 ∈ (ℤ_≥‘0) → (0..^(3 + 1)) = ((0..^3) ∪ {3}))
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (0..^(3 + 1)) = ((0..^3) ∪ {3})
40	34, 39	eqtr3i 2806	. . . . . . 7 ⊢ (0..^4) = ((0..^3) ∪ {3})
41	32, 40	syl6eq 2832	. . . . . 6 ⊢ (𝜑 → dom ⟨“𝐴𝐵𝐶𝐷”⟩ = ((0..^3) ∪ {3}))
42	41	raleqdv 3357	. . . . 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 4938	. . . . . . . 8 ⊢ (𝑗 = 3 → (𝑖 < 𝑗 ↔ 𝑖 < 3))
44		fveq2 6504	. . . . . . . . . 10 ⊢ (𝑗 = 3 → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗) = (⟨“𝐴𝐵𝐶𝐷”⟩‘3))
45	44	oveq2d 6998	. . . . . . . . 9 ⊢ (𝑗 = 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)))
46		fveq2 6504	. . . . . . . . . 10 ⊢ (𝑗 = 3 → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗) = (⟨“𝑊𝑋𝑌𝑍”⟩‘3))
47	46	oveq2d 6998	. . . . . . . . 9 ⊢ (𝑗 = 3 → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))
48	45, 47	eqeq12d 2795	. . . . . . . 8 ⊢ (𝑗 = 3 → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3))))
49	43, 48	imbi12d 337	. . . . . . 7 ⊢ (𝑗 = 3 → ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
50	49	ralunsn 4703	. . . . . 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))))))
51	35, 50	ax-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)))))
52	42, 51	syl6bb 279	. . . 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))))))
53	52	ralbidv 3149	. . 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))))))
54	41	raleqdv 3357	. . . 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 12939	. . . . . . . . . . . . . . . 16 ⊢ (0..^3) ⊆ ℕ₀
56	55, 6	sstri 3869	. . . . . . . . . . . . . . 15 ⊢ (0..^3) ⊆ ℝ
57		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 ∈ (0..^3))
58	56, 57	sseldi 3858	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 ∈ ℝ)
59		simpl 475	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑖 = 3)
60		3re 11526	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℝ
61	59, 60	syl6eqel 2876	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑖 ∈ ℝ)
62		elfzolt2 12869	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0..^3) → 𝑗 < 3)
63	62	adantl 474	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 < 3)
64	63, 59	breqtrrd 4962	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → 𝑗 < 𝑖)
65	58, 61, 64	ltnsymd 10595	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → ¬ 𝑖 < 𝑗)
66	65	pm2.21d 119	. . . . . . . . . . . 12 ⊢ ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → (𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))))
67		tbtru 1516	. . . . . . . . . . . 12 ⊢ ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⊤))
68	66, 67	sylib 210	. . . . . . . . . . 11 ⊢ ((𝑖 = 3 ∧ 𝑗 ∈ (0..^3)) → ((𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⊤))
69	68	ralbidva 3148	. . . . . . . . . 10 ⊢ (𝑖 = 3 → (∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ∀𝑗 ∈ (0..^3)⊤))
70		3nn 11525	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ
71		lbfzo0 12898	. . . . . . . . . . . . 13 ⊢ (0 ∈ (0..^3) ↔ 3 ∈ ℕ)
72	70, 71	mpbir 223	. . . . . . . . . . . 12 ⊢ 0 ∈ (0..^3)
73	72	ne0ii 4192	. . . . . . . . . . 11 ⊢ (0..^3) ≠ ∅
74		r19.3rzv 4330	. . . . . . . . . . 11 ⊢ ((0..^3) ≠ ∅ → (⊤ ↔ ∀𝑗 ∈ (0..^3)⊤))
75	73, 74	ax-mp 5	. . . . . . . . . 10 ⊢ (⊤ ↔ ∀𝑗 ∈ (0..^3)⊤)
76	69, 75	syl6bbr 281	. . . . . . . . 9 ⊢ (𝑖 = 3 → (∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⊤))
77		breq1 4937	. . . . . . . . . . . 12 ⊢ (𝑖 = 3 → (𝑖 < 3 ↔ 3 < 3))
78	60	ltnri 10555	. . . . . . . . . . . . 13 ⊢ ¬ 3 < 3
79	78	bifal 1524	. . . . . . . . . . . 12 ⊢ (3 < 3 ↔ ⊥)
80	77, 79	syl6bb 279	. . . . . . . . . . 11 ⊢ (𝑖 = 3 → (𝑖 < 3 ↔ ⊥))
81	80	imbi1d 334	. . . . . . . . . 10 ⊢ (𝑖 = 3 → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (⊥ → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
82		falim 1525	. . . . . . . . . . 11 ⊢ (⊥ → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))
83	82	bitru 1517	. . . . . . . . . 10 ⊢ ((⊥ → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ ⊤)
84	81, 83	syl6bb 279	. . . . . . . . 9 ⊢ (𝑖 = 3 → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ ⊤))
85	76, 84	anbi12d 622	. . . . . . . 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 557	. . . . . . . 8 ⊢ ((⊤ ∧ ⊤) ↔ ⊤)
87	85, 86	syl6bb 279	. . . . . . 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)))) ↔ ⊤))
88	87	ralunsn 4703	. . . . . 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)))) ∧ ⊤)))
89	35, 88	ax-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 453	. . . . 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 1519	. . . . 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)))))
92	89, 90, 91	3bitri 289	. . . 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)))))
93	54, 92	syl6bb 279	. . 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))))))
94	53, 93	bitrd 271	. 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 3122	. . 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)))))
96	9, 10, 11	s3cld 14102	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
97		wrdf 13683	. . . . . . . . 9 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 → ⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
98	96, 97	syl 17	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
99		s3len 14124	. . . . . . . . . 10 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
100	99	oveq2i 6993	. . . . . . . . 9 ⊢ (0..^(♯‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3)
101	100	feq2i 6341	. . . . . . . 8 ⊢ (⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃 ↔ ⟨“𝐴𝐵𝐶”⟩:(0..^3)⟶𝑃)
102	98, 101	sylib 210	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩:(0..^3)⟶𝑃)
103	102	fdmd 6358	. . . . . 6 ⊢ (𝜑 → dom ⟨“𝐴𝐵𝐶”⟩ = (0..^3))
104		fdm 6357	. . . . . . 7 ⊢ (⟨“𝐴𝐵𝐶”⟩:(0..^3)⟶𝑃 → dom ⟨“𝐴𝐵𝐶”⟩ = (0..^3))
105		raleq 3347	. . . . . . 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)))))
106	102, 104, 105	3syl 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)))))
107	103, 106	raleqbidv 3343	. . . . 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)))))
108	56	a1i 11	. . . . . 6 ⊢ (𝜑 → (0..^3) ⊆ ℝ)
109	20, 21, 22	s3cld 14102	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃)
110		wrdf 13683	. . . . . . . 8 ⊢ (⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃 → ⟨“𝑊𝑋𝑌”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌”⟩))⟶𝑃)
111	109, 110	syl 17	. . . . . . 7 ⊢ (𝜑 → ⟨“𝑊𝑋𝑌”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌”⟩))⟶𝑃)
112		s3len 14124	. . . . . . . . 9 ⊢ (♯‘⟨“𝑊𝑋𝑌”⟩) = 3
113	112	oveq2i 6993	. . . . . . . 8 ⊢ (0..^(♯‘⟨“𝑊𝑋𝑌”⟩)) = (0..^3)
114	113	feq2i 6341	. . . . . . 7 ⊢ (⟨“𝑊𝑋𝑌”⟩:(0..^(♯‘⟨“𝑊𝑋𝑌”⟩))⟶𝑃 ↔ ⟨“𝑊𝑋𝑌”⟩:(0..^3)⟶𝑃)
115	111, 114	sylib 210	. . . . . 6 ⊢ (𝜑 → ⟨“𝑊𝑋𝑌”⟩:(0..^3)⟶𝑃)
116	1, 2, 3, 4, 108, 102, 115	iscgrglt 26017	. . . . 5 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝑊𝑋𝑌”⟩ ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) − (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) − (⟨“𝑊𝑋𝑌”⟩‘𝑗)))))
117		df-s4 14080	. . . . . . . . . . 11 ⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
118	117	fveq1i 6505	. . . . . . . . . 10 ⊢ (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑖)
119	9	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐴 ∈ 𝑃)
120	10	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐵 ∈ 𝑃)
121	11	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐶 ∈ 𝑃)
122	119, 120, 121	s3cld 14102	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
123	12	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝐷 ∈ 𝑃)
124	123	s1cld 13772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝐷”⟩ ∈ Word 𝑃)
125		simprl 759	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑖 ∈ (0..^3))
126	125, 100	syl6eleqr 2879	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑖 ∈ (0..^(♯‘⟨“𝐴𝐵𝐶”⟩)))
127		ccatval1 13746	. . . . . . . . . . 11 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ ⟨“𝐷”⟩ ∈ Word 𝑃 ∧ 𝑖 ∈ (0..^(♯‘⟨“𝐴𝐵𝐶”⟩))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
128	122, 124, 126, 127	syl3anc 1352	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
129	118, 128	syl5eq 2828	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
130	117	fveq1i 6505	. . . . . . . . . 10 ⊢ (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗) = ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑗)
131		simprr 761	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑗 ∈ (0..^3))
132	131, 100	syl6eleqr 2879	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑗 ∈ (0..^(♯‘⟨“𝐴𝐵𝐶”⟩)))
133		ccatval1 13746	. . . . . . . . . . 11 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ ⟨“𝐷”⟩ ∈ Word 𝑃 ∧ 𝑗 ∈ (0..^(♯‘⟨“𝐴𝐵𝐶”⟩))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘𝑗))
134	122, 124, 132, 133	syl3anc 1352	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘𝑗))
135	130, 134	syl5eq 2828	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘𝑗))
136	129, 135	oveq12d 7000	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) − (⟨“𝐴𝐵𝐶”⟩‘𝑗)))
137		df-s4 14080	. . . . . . . . . . 11 ⊢ ⟨“𝑊𝑋𝑌𝑍”⟩ = (⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)
138	137	fveq1i 6505	. . . . . . . . . 10 ⊢ (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑖)
139	20	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑊 ∈ 𝑃)
140	21	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑋 ∈ 𝑃)
141	22	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑌 ∈ 𝑃)
142	139, 140, 141	s3cld 14102	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃)
143	23	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑍 ∈ 𝑃)
144	143	s1cld 13772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ⟨“𝑍”⟩ ∈ Word 𝑃)
145	125, 113	syl6eleqr 2879	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑖 ∈ (0..^(♯‘⟨“𝑊𝑋𝑌”⟩)))
146		ccatval1 13746	. . . . . . . . . . 11 ⊢ ((⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃 ∧ ⟨“𝑍”⟩ ∈ Word 𝑃 ∧ 𝑖 ∈ (0..^(♯‘⟨“𝑊𝑋𝑌”⟩))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑖) = (⟨“𝑊𝑋𝑌”⟩‘𝑖))
147	142, 144, 145, 146	syl3anc 1352	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑖) = (⟨“𝑊𝑋𝑌”⟩‘𝑖))
148	138, 147	syl5eq 2828	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌”⟩‘𝑖))
149	137	fveq1i 6505	. . . . . . . . . 10 ⊢ (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗) = ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑗)
150	131, 113	syl6eleqr 2879	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → 𝑗 ∈ (0..^(♯‘⟨“𝑊𝑋𝑌”⟩)))
151		ccatval1 13746	. . . . . . . . . . 11 ⊢ ((⟨“𝑊𝑋𝑌”⟩ ∈ Word 𝑃 ∧ ⟨“𝑍”⟩ ∈ Word 𝑃 ∧ 𝑗 ∈ (0..^(♯‘⟨“𝑊𝑋𝑌”⟩))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑗) = (⟨“𝑊𝑋𝑌”⟩‘𝑗))
152	142, 144, 150, 151	syl3anc 1352	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝑊𝑋𝑌”⟩ ++ ⟨“𝑍”⟩)‘𝑗) = (⟨“𝑊𝑋𝑌”⟩‘𝑗))
153	149, 152	syl5eq 2828	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗) = (⟨“𝑊𝑋𝑌”⟩‘𝑗))
154	148, 153	oveq12d 7000	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) − (⟨“𝑊𝑋𝑌”⟩‘𝑗)))
155	136, 154	eqeq12d 2795	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^3) ∧ 𝑗 ∈ (0..^3))) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) − (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌”⟩‘𝑖) − (⟨“𝑊𝑋𝑌”⟩‘𝑗))))
156	155	imbi2d 333	. . . . . 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)))))
157	156	2ralbidva 3150	. . . . 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)))))
158	107, 116, 157	3bitr4rd 304	. . . 4 ⊢ (𝜑 → (∀𝑖 ∈ (0..^3)∀𝑗 ∈ (0..^3)(𝑖 < 𝑗 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑗)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑗))) ↔ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝑊𝑋𝑌”⟩))
159		fzo0to3tp 12944	. . . . . 6 ⊢ (0..^3) = {0, 1, 2}
160		raleq 3347	. . . . . 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)))))
161	159, 160	mp1i 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)))))
162		3pos 11558	. . . . . . . . . 10 ⊢ 0 < 3
163		breq1 4937	. . . . . . . . . 10 ⊢ (𝑖 = 0 → (𝑖 < 3 ↔ 0 < 3))
164	162, 163	mpbiri 250	. . . . . . . . 9 ⊢ (𝑖 = 0 → 𝑖 < 3)
165	164	adantl 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 0) → 𝑖 < 3)
166		biimt 353	. . . . . . . 8 ⊢ (𝑖 < 3 → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
167	165, 166	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
168		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 = 0) → 𝑖 = 0)
169	168	fveq2d 6508	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 0) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶𝐷”⟩‘0))
170		s4fv0 14125	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
171	9, 170	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
172	171	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 0) → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)
173	169, 172	eqtrd 2816	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 0) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = 𝐴)
174		s4fv3 14128	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
175	12, 174	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
176	175	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 0) → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
177	173, 176	oveq12d 7000	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = (𝐴 − 𝐷))
178	168	fveq2d 6508	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 0) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌𝑍”⟩‘0))
179		s4fv0 14125	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ 𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘0) = 𝑊)
180	20, 179	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘0) = 𝑊)
181	180	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 0) → (⟨“𝑊𝑋𝑌𝑍”⟩‘0) = 𝑊)
182	178, 181	eqtrd 2816	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 0) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = 𝑊)
183		s4fv3 14128	. . . . . . . . . . 11 ⊢ (𝑍 ∈ 𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
184	23, 183	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
185	184	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 0) → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
186	182, 185	oveq12d 7000	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) = (𝑊 − 𝑍))
187	177, 186	eqeq12d 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝐴 − 𝐷) = (𝑊 − 𝑍)))
188	167, 187	bitr3d 273	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (𝐴 − 𝐷) = (𝑊 − 𝑍)))
189		1lt3 11626	. . . . . . . . . 10 ⊢ 1 < 3
190		breq1 4937	. . . . . . . . . 10 ⊢ (𝑖 = 1 → (𝑖 < 3 ↔ 1 < 3))
191	189, 190	mpbiri 250	. . . . . . . . 9 ⊢ (𝑖 = 1 → 𝑖 < 3)
192	191	adantl 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 1) → 𝑖 < 3)
193	192, 166	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 1) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
194		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 = 1) → 𝑖 = 1)
195	194	fveq2d 6508	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 1) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶𝐷”⟩‘1))
196		s4fv1 14126	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
197	10, 196	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
198	197	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 1) → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)
199	195, 198	eqtrd 2816	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 1) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = 𝐵)
200	175	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 1) → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
201	199, 200	oveq12d 7000	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 1) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = (𝐵 − 𝐷))
202	194	fveq2d 6508	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 1) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌𝑍”⟩‘1))
203		s4fv1 14126	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘1) = 𝑋)
204	21, 203	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘1) = 𝑋)
205	204	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 1) → (⟨“𝑊𝑋𝑌𝑍”⟩‘1) = 𝑋)
206	202, 205	eqtrd 2816	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 1) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = 𝑋)
207	184	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 1) → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
208	206, 207	oveq12d 7000	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 1) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) = (𝑋 − 𝑍))
209	201, 208	eqeq12d 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 1) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝐵 − 𝐷) = (𝑋 − 𝑍)))
210	193, 209	bitr3d 273	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 1) → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (𝐵 − 𝐷) = (𝑋 − 𝑍)))
211		2lt3 11625	. . . . . . . . . 10 ⊢ 2 < 3
212		breq1 4937	. . . . . . . . . 10 ⊢ (𝑖 = 2 → (𝑖 < 3 ↔ 2 < 3))
213	211, 212	mpbiri 250	. . . . . . . . 9 ⊢ (𝑖 = 2 → 𝑖 < 3)
214	213	adantl 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 2) → 𝑖 < 3)
215	214, 166	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 2) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)))))
216		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 = 2) → 𝑖 = 2)
217	216	fveq2d 6508	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 2) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = (⟨“𝐴𝐵𝐶𝐷”⟩‘2))
218		s4fv2 14127	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝑃 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
219	11, 218	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
220	219	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 2) → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)
221	217, 220	eqtrd 2816	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 2) → (⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) = 𝐶)
222	175	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 2) → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)
223	221, 222	oveq12d 7000	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 2) → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = (𝐶 − 𝐷))
224	216	fveq2d 6508	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 2) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = (⟨“𝑊𝑋𝑌𝑍”⟩‘2))
225		s4fv2 14127	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ 𝑃 → (⟨“𝑊𝑋𝑌𝑍”⟩‘2) = 𝑌)
226	22, 225	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨“𝑊𝑋𝑌𝑍”⟩‘2) = 𝑌)
227	226	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 2) → (⟨“𝑊𝑋𝑌𝑍”⟩‘2) = 𝑌)
228	224, 227	eqtrd 2816	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 2) → (⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) = 𝑌)
229	184	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 2) → (⟨“𝑊𝑋𝑌𝑍”⟩‘3) = 𝑍)
230	228, 229	oveq12d 7000	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 2) → ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) = (𝑌 − 𝑍))
231	223, 230	eqeq12d 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 2) → (((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3)) ↔ (𝐶 − 𝐷) = (𝑌 − 𝑍)))
232	215, 231	bitr3d 273	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 2) → ((𝑖 < 3 → ((⟨“𝐴𝐵𝐶𝐷”⟩‘𝑖) − (⟨“𝐴𝐵𝐶𝐷”⟩‘3)) = ((⟨“𝑊𝑋𝑌𝑍”⟩‘𝑖) − (⟨“𝑊𝑋𝑌𝑍”⟩‘3))) ↔ (𝐶 − 𝐷) = (𝑌 − 𝑍)))
233		0red 10449	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
234		1red 10446	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
235		2re 11520	. . . . . . 7 ⊢ 2 ∈ ℝ
236	235	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
237	188, 210, 232, 233, 234, 236	raltpd 4595	. . . . 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))))
238	161, 237	bitrd 271	. . . 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))))
239	158, 238	anbi12d 622	. . 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)))))
240	95, 239	syl5bb 275	. 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)))))
241	31, 94, 240	3bitrd 297	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 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ⊤wtru 1509 ⊥wfal 1520 ∈ wcel 2051 ≠ wne 2969 ∀wral 3090 ∪ cun 3829 ⊆ wss 3831 ∅c0 4181 {csn 4444 {ctp 4448 class class class wbr 4934 dom cdm 5411 ⟶wf 6189 ‘cfv 6193 (class class class)co 6982 ℝcr 10340 0cc0 10341 1c1 10342 + caddc 10344 < clt 10480 ℕcn 11445 2c2 11501 3c3 11502 4c4 11503 ℕ₀cn0 11713 ℤ_≥cuz 12064 ..^cfzo 12855 ♯chash 13511 Word cword 13678 ++ cconcat 13739 ⟨“cs1 13764 ⟨“cs3 14072 ⟨“cs4 14073 Basecbs 16345 distcds 16436 TarskiGcstrkg 25933 Itvcitv 25939 cgrGccgrg 26013

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-1st 7507 df-2nd 7508 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-oadd 7915 df-er 8095 df-pm 8215 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-card 9168 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-nn 11446 df-2 11509 df-3 11510 df-4 11511 df-n0 11714 df-z 11800 df-uz 12065 df-fz 12715 df-fzo 12856 df-hash 13512 df-word 13679 df-concat 13740 df-s1 13765 df-s2 14078 df-s3 14079 df-s4 14080 df-trkgc 25951 df-trkgcb 25953 df-trkg 25956 df-cgrg 26014

This theorem is referenced by: cgrg3col4 26357

W3C validator