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Theorem trgcgrg 27457
Description: The property for two triangles to be congruent to each other. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
trgcgrg.p 𝑃 = (Base‘𝐺)
trgcgrg.m = (dist‘𝐺)
trgcgrg.r = (cgrG‘𝐺)
trgcgrg.g (𝜑𝐺 ∈ TarskiG)
trgcgrg.a (𝜑𝐴𝑃)
trgcgrg.b (𝜑𝐵𝑃)
trgcgrg.c (𝜑𝐶𝑃)
trgcgrg.d (𝜑𝐷𝑃)
trgcgrg.e (𝜑𝐸𝑃)
trgcgrg.f (𝜑𝐹𝑃)
Assertion
Ref Expression
trgcgrg (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))))

Proof of Theorem trgcgrg
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trgcgrg.a . . . . . . 7 (𝜑𝐴𝑃)
2 trgcgrg.b . . . . . . 7 (𝜑𝐵𝑃)
3 trgcgrg.c . . . . . . 7 (𝜑𝐶𝑃)
41, 2, 3s3cld 14761 . . . . . 6 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
5 wrdf 14407 . . . . . 6 (⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 → ⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
64, 5syl 17 . . . . 5 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
7 s3len 14783 . . . . . . . 8 (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
87oveq2i 7368 . . . . . . 7 (0..^(♯‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3)
9 fzo0to3tp 13658 . . . . . . 7 (0..^3) = {0, 1, 2}
108, 9eqtri 2764 . . . . . 6 (0..^(♯‘⟨“𝐴𝐵𝐶”⟩)) = {0, 1, 2}
1110feq2i 6660 . . . . 5 (⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃 ↔ ⟨“𝐴𝐵𝐶”⟩:{0, 1, 2}⟶𝑃)
126, 11sylib 217 . . . 4 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:{0, 1, 2}⟶𝑃)
1312fdmd 6679 . . 3 (𝜑 → dom ⟨“𝐴𝐵𝐶”⟩ = {0, 1, 2})
1413raleqdv 3313 . . 3 (𝜑 → (∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗))))
1513, 14raleqbidv 3319 . 2 (𝜑 → (∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ∀𝑖 ∈ {0, 1, 2}∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗))))
16 trgcgrg.p . . 3 𝑃 = (Base‘𝐺)
17 trgcgrg.m . . 3 = (dist‘𝐺)
18 trgcgrg.r . . 3 = (cgrG‘𝐺)
19 trgcgrg.g . . 3 (𝜑𝐺 ∈ TarskiG)
20 0re 11157 . . . . 5 0 ∈ ℝ
21 1re 11155 . . . . 5 1 ∈ ℝ
22 2re 12227 . . . . 5 2 ∈ ℝ
23 tpssi 4796 . . . . 5 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 2 ∈ ℝ) → {0, 1, 2} ⊆ ℝ)
2420, 21, 22, 23mp3an 1461 . . . 4 {0, 1, 2} ⊆ ℝ
2524a1i 11 . . 3 (𝜑 → {0, 1, 2} ⊆ ℝ)
26 trgcgrg.d . . . . . 6 (𝜑𝐷𝑃)
27 trgcgrg.e . . . . . 6 (𝜑𝐸𝑃)
28 trgcgrg.f . . . . . 6 (𝜑𝐹𝑃)
2926, 27, 28s3cld 14761 . . . . 5 (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃)
30 wrdf 14407 . . . . 5 (⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃 → ⟨“𝐷𝐸𝐹”⟩:(0..^(♯‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃)
3129, 30syl 17 . . . 4 (𝜑 → ⟨“𝐷𝐸𝐹”⟩:(0..^(♯‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃)
32 s3len 14783 . . . . . . 7 (♯‘⟨“𝐷𝐸𝐹”⟩) = 3
3332oveq2i 7368 . . . . . 6 (0..^(♯‘⟨“𝐷𝐸𝐹”⟩)) = (0..^3)
3433, 9eqtri 2764 . . . . 5 (0..^(♯‘⟨“𝐷𝐸𝐹”⟩)) = {0, 1, 2}
3534feq2i 6660 . . . 4 (⟨“𝐷𝐸𝐹”⟩:(0..^(♯‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃 ↔ ⟨“𝐷𝐸𝐹”⟩:{0, 1, 2}⟶𝑃)
3631, 35sylib 217 . . 3 (𝜑 → ⟨“𝐷𝐸𝐹”⟩:{0, 1, 2}⟶𝑃)
3716, 17, 18, 19, 25, 12, 36iscgrgd 27455 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩ ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗))))
38 fveq2 6842 . . . . . . . . . 10 (𝑗 = 0 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘0))
39 s3fv0 14780 . . . . . . . . . . 11 (𝐴𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
401, 39syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
4138, 40sylan9eqr 2798 . . . . . . . . 9 ((𝜑𝑗 = 0) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐴)
4241oveq2d 7373 . . . . . . . 8 ((𝜑𝑗 = 0) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
43 fveq2 6842 . . . . . . . . . 10 (𝑗 = 0 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘0))
44 s3fv0 14780 . . . . . . . . . . 11 (𝐷𝑃 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
4526, 44syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
4643, 45sylan9eqr 2798 . . . . . . . . 9 ((𝜑𝑗 = 0) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐷)
4746oveq2d 7373 . . . . . . . 8 ((𝜑𝑗 = 0) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
4842, 47eqeq12d 2752 . . . . . . 7 ((𝜑𝑗 = 0) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
49 fveq2 6842 . . . . . . . . . 10 (𝑗 = 1 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘1))
50 s3fv1 14781 . . . . . . . . . . 11 (𝐵𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
512, 50syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
5249, 51sylan9eqr 2798 . . . . . . . . 9 ((𝜑𝑗 = 1) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐵)
5352oveq2d 7373 . . . . . . . 8 ((𝜑𝑗 = 1) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
54 fveq2 6842 . . . . . . . . . 10 (𝑗 = 1 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘1))
55 s3fv1 14781 . . . . . . . . . . 11 (𝐸𝑃 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
5627, 55syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
5754, 56sylan9eqr 2798 . . . . . . . . 9 ((𝜑𝑗 = 1) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐸)
5857oveq2d 7373 . . . . . . . 8 ((𝜑𝑗 = 1) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
5953, 58eqeq12d 2752 . . . . . . 7 ((𝜑𝑗 = 1) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
60 fveq2 6842 . . . . . . . . . 10 (𝑗 = 2 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘2))
61 s3fv2 14782 . . . . . . . . . . 11 (𝐶𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
623, 61syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
6360, 62sylan9eqr 2798 . . . . . . . . 9 ((𝜑𝑗 = 2) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐶)
6463oveq2d 7373 . . . . . . . 8 ((𝜑𝑗 = 2) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
65 fveq2 6842 . . . . . . . . . 10 (𝑗 = 2 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘2))
66 s3fv2 14782 . . . . . . . . . . 11 (𝐹𝑃 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
6728, 66syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
6865, 67sylan9eqr 2798 . . . . . . . . 9 ((𝜑𝑗 = 2) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐹)
6968oveq2d 7373 . . . . . . . 8 ((𝜑𝑗 = 2) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
7064, 69eqeq12d 2752 . . . . . . 7 ((𝜑𝑗 = 2) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
71 0red 11158 . . . . . . 7 (𝜑 → 0 ∈ ℝ)
72 1red 11156 . . . . . . 7 (𝜑 → 1 ∈ ℝ)
7322a1i 11 . . . . . . 7 (𝜑 → 2 ∈ ℝ)
7448, 59, 70, 71, 72, 73raltpd 4742 . . . . . 6 (𝜑 → (∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ (((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))))
7574adantr 481 . . . . 5 ((𝜑𝑖 = 0) → (∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ (((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))))
76 fveq2 6842 . . . . . . . . . 10 (𝑖 = 0 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘0))
7776adantl 482 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘0))
7840adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
7977, 78eqtr2d 2777 . . . . . . . 8 ((𝜑𝑖 = 0) → 𝐴 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
8079oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
81 fveq2 6842 . . . . . . . . . 10 (𝑖 = 0 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘0))
8281adantl 482 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘0))
8345adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
8482, 83eqtr2d 2777 . . . . . . . 8 ((𝜑𝑖 = 0) → 𝐷 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
8584oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
8680, 85eqeq12d 2752 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐴) = (𝐷 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
8779oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
8884oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
8987, 88eqeq12d 2752 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐵) = (𝐷 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
9079oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
9184oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
9290, 91eqeq12d 2752 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐶) = (𝐷 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
9386, 89, 923anbi123d 1436 . . . . 5 ((𝜑𝑖 = 0) → (((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐴 𝐶) = (𝐷 𝐹)) ↔ (((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))))
9475, 93bitr4d 281 . . . 4 ((𝜑𝑖 = 0) → (∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐴 𝐶) = (𝐷 𝐹))))
9574adantr 481 . . . . 5 ((𝜑𝑖 = 1) → (∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ (((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))))
96 fveq2 6842 . . . . . . . . . 10 (𝑖 = 1 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘1))
9796adantl 482 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘1))
9851adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
9997, 98eqtr2d 2777 . . . . . . . 8 ((𝜑𝑖 = 1) → 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
10099oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
101 fveq2 6842 . . . . . . . . . 10 (𝑖 = 1 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘1))
102101adantl 482 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘1))
10356adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
104102, 103eqtr2d 2777 . . . . . . . 8 ((𝜑𝑖 = 1) → 𝐸 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
105104oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
106100, 105eqeq12d 2752 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐴) = (𝐸 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
10799oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
108104oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
109107, 108eqeq12d 2752 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐵) = (𝐸 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
11099oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
111104oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
112110, 111eqeq12d 2752 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐶) = (𝐸 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
113106, 109, 1123anbi123d 1436 . . . . 5 ((𝜑𝑖 = 1) → (((𝐵 𝐴) = (𝐸 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹)) ↔ (((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))))
11495, 113bitr4d 281 . . . 4 ((𝜑𝑖 = 1) → (∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((𝐵 𝐴) = (𝐸 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹))))
11574adantr 481 . . . . 5 ((𝜑𝑖 = 2) → (∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ (((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))))
116 fveq2 6842 . . . . . . . . . 10 (𝑖 = 2 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘2))
117116adantl 482 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘2))
11862adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
119117, 118eqtr2d 2777 . . . . . . . 8 ((𝜑𝑖 = 2) → 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
120119oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
121 fveq2 6842 . . . . . . . . . 10 (𝑖 = 2 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘2))
122121adantl 482 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘2))
12367adantr 481 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
124122, 123eqtr2d 2777 . . . . . . . 8 ((𝜑𝑖 = 2) → 𝐹 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
125124oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
126120, 125eqeq12d 2752 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐴) = (𝐹 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
127119oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
128124oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
129127, 128eqeq12d 2752 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐵) = (𝐹 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
130119oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
131124oveq1d 7372 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
132130, 131eqeq12d 2752 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐶) = (𝐹 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
133126, 129, 1323anbi123d 1436 . . . . 5 ((𝜑𝑖 = 2) → (((𝐶 𝐴) = (𝐹 𝐷) ∧ (𝐶 𝐵) = (𝐹 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹)) ↔ (((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸) ∧ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))))
134115, 133bitr4d 281 . . . 4 ((𝜑𝑖 = 2) → (∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((𝐶 𝐴) = (𝐹 𝐷) ∧ (𝐶 𝐵) = (𝐹 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹))))
13594, 114, 134, 71, 72, 73raltpd 4742 . . 3 (𝜑 → (∀𝑖 ∈ {0, 1, 2}∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ (((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐴 𝐶) = (𝐷 𝐹)) ∧ ((𝐵 𝐴) = (𝐸 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹)) ∧ ((𝐶 𝐴) = (𝐹 𝐷) ∧ (𝐶 𝐵) = (𝐹 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹)))))
136 an33rean 1483 . . . 4 ((((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐴 𝐶) = (𝐷 𝐹)) ∧ ((𝐵 𝐴) = (𝐸 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹)) ∧ ((𝐶 𝐴) = (𝐹 𝐷) ∧ (𝐶 𝐵) = (𝐹 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹))) ↔ (((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹)) ∧ (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)) ∧ ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)) ∧ ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)))))
137 eqid 2736 . . . . . . . 8 (Itv‘𝐺) = (Itv‘𝐺)
13816, 17, 137, 19, 1, 26tgcgrtriv 27426 . . . . . . 7 (𝜑 → (𝐴 𝐴) = (𝐷 𝐷))
13916, 17, 137, 19, 2, 27tgcgrtriv 27426 . . . . . . 7 (𝜑 → (𝐵 𝐵) = (𝐸 𝐸))
14016, 17, 137, 19, 3, 28tgcgrtriv 27426 . . . . . . 7 (𝜑 → (𝐶 𝐶) = (𝐹 𝐹))
141138, 139, 1403jca 1128 . . . . . 6 (𝜑 → ((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹)))
142141biantrurd 533 . . . . 5 (𝜑 → ((((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)) ∧ ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)) ∧ ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))) ↔ (((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹)) ∧ (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)) ∧ ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)) ∧ ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))))))
143 simprl 769 . . . . . . 7 ((𝜑 ∧ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷))) → (𝐴 𝐵) = (𝐷 𝐸))
144 simpr 485 . . . . . . . 8 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → (𝐴 𝐵) = (𝐷 𝐸))
14519adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐺 ∈ TarskiG)
1461adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐴𝑃)
1472adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐵𝑃)
14826adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐷𝑃)
14927adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐸𝑃)
15016, 17, 137, 145, 146, 147, 148, 149, 144tgcgrcomlr 27422 . . . . . . . 8 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → (𝐵 𝐴) = (𝐸 𝐷))
151144, 150jca 512 . . . . . . 7 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)))
152143, 151impbida 799 . . . . . 6 (𝜑 → (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)) ↔ (𝐴 𝐵) = (𝐷 𝐸)))
153 simprl 769 . . . . . . 7 ((𝜑 ∧ ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸))) → (𝐵 𝐶) = (𝐸 𝐹))
154 simpr 485 . . . . . . . 8 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → (𝐵 𝐶) = (𝐸 𝐹))
15519adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐺 ∈ TarskiG)
1562adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐵𝑃)
1573adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐶𝑃)
15827adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐸𝑃)
15928adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐹𝑃)
16016, 17, 137, 155, 156, 157, 158, 159, 154tgcgrcomlr 27422 . . . . . . . 8 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → (𝐶 𝐵) = (𝐹 𝐸))
161154, 160jca 512 . . . . . . 7 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)))
162153, 161impbida 799 . . . . . 6 (𝜑 → (((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)) ↔ (𝐵 𝐶) = (𝐸 𝐹)))
163 simprr 771 . . . . . . 7 ((𝜑 ∧ ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))) → (𝐶 𝐴) = (𝐹 𝐷))
16419adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐺 ∈ TarskiG)
1653adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐶𝑃)
1661adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐴𝑃)
16728adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐹𝑃)
16826adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐷𝑃)
169 simpr 485 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → (𝐶 𝐴) = (𝐹 𝐷))
17016, 17, 137, 164, 165, 166, 167, 168, 169tgcgrcomlr 27422 . . . . . . . 8 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → (𝐴 𝐶) = (𝐷 𝐹))
171170, 169jca 512 . . . . . . 7 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)))
172163, 171impbida 799 . . . . . 6 (𝜑 → (((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)) ↔ (𝐶 𝐴) = (𝐹 𝐷)))
173152, 162, 1723anbi123d 1436 . . . . 5 (𝜑 → ((((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)) ∧ ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)) ∧ ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))) ↔ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))))
174142, 173bitr3d 280 . . . 4 (𝜑 → ((((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹)) ∧ (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)) ∧ ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)) ∧ ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)))) ↔ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))))
175136, 174bitrid 282 . . 3 (𝜑 → ((((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐴 𝐶) = (𝐷 𝐹)) ∧ ((𝐵 𝐴) = (𝐸 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹)) ∧ ((𝐶 𝐴) = (𝐹 𝐷) ∧ (𝐶 𝐵) = (𝐹 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹))) ↔ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))))
176135, 175bitr2d 279 . 2 (𝜑 → (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)) ↔ ∀𝑖 ∈ {0, 1, 2}∀𝑗 ∈ {0, 1, 2} ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗))))
17715, 37, 1763bitr4d 310 1 (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩ ↔ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wss 3910  {ctp 4590   class class class wbr 5105  dom cdm 5633  wf 6492  cfv 6496  (class class class)co 7357  cr 11050  0cc0 11051  1c1 11052  2c2 12208  3c3 12209  ..^cfzo 13567  chash 14230  Word cword 14402  ⟨“cs3 14731  Basecbs 17083  distcds 17142  TarskiGcstrkg 27369  Itvcitv 27375  cgrGccgrg 27452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-s2 14737  df-s3 14738  df-trkgc 27390  df-trkgcb 27392  df-trkg 27395  df-cgrg 27453
This theorem is referenced by:  trgcgr  27458  cgr3simp1  27462  cgr3simp2  27463  cgr3simp3  27464  dfcgrg2  27805
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