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Theorem trgcgrg 26606
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 14437 . . . . . 6 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
5 wrdf 14074 . . . . . 6 (⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 → ⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
64, 5syl 17 . . . . 5 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃)
7 s3len 14459 . . . . . . . 8 (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
87oveq2i 7224 . . . . . . 7 (0..^(♯‘⟨“𝐴𝐵𝐶”⟩)) = (0..^3)
9 fzo0to3tp 13328 . . . . . . 7 (0..^3) = {0, 1, 2}
108, 9eqtri 2765 . . . . . 6 (0..^(♯‘⟨“𝐴𝐵𝐶”⟩)) = {0, 1, 2}
1110feq2i 6537 . . . . 5 (⟨“𝐴𝐵𝐶”⟩:(0..^(♯‘⟨“𝐴𝐵𝐶”⟩))⟶𝑃 ↔ ⟨“𝐴𝐵𝐶”⟩:{0, 1, 2}⟶𝑃)
126, 11sylib 221 . . . 4 (𝜑 → ⟨“𝐴𝐵𝐶”⟩:{0, 1, 2}⟶𝑃)
1312fdmd 6556 . . 3 (𝜑 → dom ⟨“𝐴𝐵𝐶”⟩ = {0, 1, 2})
1413raleqdv 3325 . . 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 3313 . 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 10835 . . . . 5 0 ∈ ℝ
21 1re 10833 . . . . 5 1 ∈ ℝ
22 2re 11904 . . . . 5 2 ∈ ℝ
23 tpssi 4749 . . . . 5 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 2 ∈ ℝ) → {0, 1, 2} ⊆ ℝ)
2420, 21, 22, 23mp3an 1463 . . . 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 14437 . . . . 5 (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃)
30 wrdf 14074 . . . . 5 (⟨“𝐷𝐸𝐹”⟩ ∈ Word 𝑃 → ⟨“𝐷𝐸𝐹”⟩:(0..^(♯‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃)
3129, 30syl 17 . . . 4 (𝜑 → ⟨“𝐷𝐸𝐹”⟩:(0..^(♯‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃)
32 s3len 14459 . . . . . . 7 (♯‘⟨“𝐷𝐸𝐹”⟩) = 3
3332oveq2i 7224 . . . . . 6 (0..^(♯‘⟨“𝐷𝐸𝐹”⟩)) = (0..^3)
3433, 9eqtri 2765 . . . . 5 (0..^(♯‘⟨“𝐷𝐸𝐹”⟩)) = {0, 1, 2}
3534feq2i 6537 . . . 4 (⟨“𝐷𝐸𝐹”⟩:(0..^(♯‘⟨“𝐷𝐸𝐹”⟩))⟶𝑃 ↔ ⟨“𝐷𝐸𝐹”⟩:{0, 1, 2}⟶𝑃)
3631, 35sylib 221 . . 3 (𝜑 → ⟨“𝐷𝐸𝐹”⟩:{0, 1, 2}⟶𝑃)
3716, 17, 18, 19, 25, 12, 36iscgrgd 26604 . 2 (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩ ↔ ∀𝑖 ∈ dom ⟨“𝐴𝐵𝐶”⟩∀𝑗 ∈ dom ⟨“𝐴𝐵𝐶”⟩((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗))))
38 fveq2 6717 . . . . . . . . . 10 (𝑗 = 0 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘0))
39 s3fv0 14456 . . . . . . . . . . 11 (𝐴𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
401, 39syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
4138, 40sylan9eqr 2800 . . . . . . . . 9 ((𝜑𝑗 = 0) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐴)
4241oveq2d 7229 . . . . . . . 8 ((𝜑𝑗 = 0) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
43 fveq2 6717 . . . . . . . . . 10 (𝑗 = 0 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘0))
44 s3fv0 14456 . . . . . . . . . . 11 (𝐷𝑃 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
4526, 44syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
4643, 45sylan9eqr 2800 . . . . . . . . 9 ((𝜑𝑗 = 0) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐷)
4746oveq2d 7229 . . . . . . . 8 ((𝜑𝑗 = 0) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
4842, 47eqeq12d 2753 . . . . . . 7 ((𝜑𝑗 = 0) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
49 fveq2 6717 . . . . . . . . . 10 (𝑗 = 1 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘1))
50 s3fv1 14457 . . . . . . . . . . 11 (𝐵𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
512, 50syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
5249, 51sylan9eqr 2800 . . . . . . . . 9 ((𝜑𝑗 = 1) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐵)
5352oveq2d 7229 . . . . . . . 8 ((𝜑𝑗 = 1) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
54 fveq2 6717 . . . . . . . . . 10 (𝑗 = 1 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘1))
55 s3fv1 14457 . . . . . . . . . . 11 (𝐸𝑃 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
5627, 55syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
5754, 56sylan9eqr 2800 . . . . . . . . 9 ((𝜑𝑗 = 1) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐸)
5857oveq2d 7229 . . . . . . . 8 ((𝜑𝑗 = 1) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
5953, 58eqeq12d 2753 . . . . . . 7 ((𝜑𝑗 = 1) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
60 fveq2 6717 . . . . . . . . . 10 (𝑗 = 2 → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = (⟨“𝐴𝐵𝐶”⟩‘2))
61 s3fv2 14458 . . . . . . . . . . 11 (𝐶𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
623, 61syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
6360, 62sylan9eqr 2800 . . . . . . . . 9 ((𝜑𝑗 = 2) → (⟨“𝐴𝐵𝐶”⟩‘𝑗) = 𝐶)
6463oveq2d 7229 . . . . . . . 8 ((𝜑𝑗 = 2) → ((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
65 fveq2 6717 . . . . . . . . . 10 (𝑗 = 2 → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = (⟨“𝐷𝐸𝐹”⟩‘2))
66 s3fv2 14458 . . . . . . . . . . 11 (𝐹𝑃 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
6728, 66syl 17 . . . . . . . . . 10 (𝜑 → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
6865, 67sylan9eqr 2800 . . . . . . . . 9 ((𝜑𝑗 = 2) → (⟨“𝐷𝐸𝐹”⟩‘𝑗) = 𝐹)
6968oveq2d 7229 . . . . . . . 8 ((𝜑𝑗 = 2) → ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
7064, 69eqeq12d 2753 . . . . . . 7 ((𝜑𝑗 = 2) → (((⟨“𝐴𝐵𝐶”⟩‘𝑖) (⟨“𝐴𝐵𝐶”⟩‘𝑗)) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) (⟨“𝐷𝐸𝐹”⟩‘𝑗)) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
71 0red 10836 . . . . . . 7 (𝜑 → 0 ∈ ℝ)
72 1red 10834 . . . . . . 7 (𝜑 → 1 ∈ ℝ)
7322a1i 11 . . . . . . 7 (𝜑 → 2 ∈ ℝ)
7448, 59, 70, 71, 72, 73raltpd 4697 . . . . . 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 484 . . . . 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 6717 . . . . . . . . . 10 (𝑖 = 0 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘0))
7776adantl 485 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘0))
7840adantr 484 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
7977, 78eqtr2d 2778 . . . . . . . 8 ((𝜑𝑖 = 0) → 𝐴 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
8079oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
81 fveq2 6717 . . . . . . . . . 10 (𝑖 = 0 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘0))
8281adantl 485 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘0))
8345adantr 484 . . . . . . . . 9 ((𝜑𝑖 = 0) → (⟨“𝐷𝐸𝐹”⟩‘0) = 𝐷)
8482, 83eqtr2d 2778 . . . . . . . 8 ((𝜑𝑖 = 0) → 𝐷 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
8584oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
8680, 85eqeq12d 2753 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐴) = (𝐷 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
8779oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
8884oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
8987, 88eqeq12d 2753 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐵) = (𝐷 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
9079oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐴 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
9184oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 0) → (𝐷 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
9290, 91eqeq12d 2753 . . . . . 6 ((𝜑𝑖 = 0) → ((𝐴 𝐶) = (𝐷 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
9386, 89, 923anbi123d 1438 . . . . 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 285 . . . 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 484 . . . . 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 6717 . . . . . . . . . 10 (𝑖 = 1 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘1))
9796adantl 485 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘1))
9851adantr 484 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
9997, 98eqtr2d 2778 . . . . . . . 8 ((𝜑𝑖 = 1) → 𝐵 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
10099oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
101 fveq2 6717 . . . . . . . . . 10 (𝑖 = 1 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘1))
102101adantl 485 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘1))
10356adantr 484 . . . . . . . . 9 ((𝜑𝑖 = 1) → (⟨“𝐷𝐸𝐹”⟩‘1) = 𝐸)
104102, 103eqtr2d 2778 . . . . . . . 8 ((𝜑𝑖 = 1) → 𝐸 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
105104oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
106100, 105eqeq12d 2753 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐴) = (𝐸 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
10799oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
108104oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
109107, 108eqeq12d 2753 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐵) = (𝐸 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
11099oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐵 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
111104oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 1) → (𝐸 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
112110, 111eqeq12d 2753 . . . . . 6 ((𝜑𝑖 = 1) → ((𝐵 𝐶) = (𝐸 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
113106, 109, 1123anbi123d 1438 . . . . 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 285 . . . 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 484 . . . . 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 6717 . . . . . . . . . 10 (𝑖 = 2 → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘2))
117116adantl 485 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶”⟩‘𝑖) = (⟨“𝐴𝐵𝐶”⟩‘2))
11862adantr 484 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
119117, 118eqtr2d 2778 . . . . . . . 8 ((𝜑𝑖 = 2) → 𝐶 = (⟨“𝐴𝐵𝐶”⟩‘𝑖))
120119oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐴) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴))
121 fveq2 6717 . . . . . . . . . 10 (𝑖 = 2 → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘2))
122121adantl 485 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐷𝐸𝐹”⟩‘𝑖) = (⟨“𝐷𝐸𝐹”⟩‘2))
12367adantr 484 . . . . . . . . 9 ((𝜑𝑖 = 2) → (⟨“𝐷𝐸𝐹”⟩‘2) = 𝐹)
124122, 123eqtr2d 2778 . . . . . . . 8 ((𝜑𝑖 = 2) → 𝐹 = (⟨“𝐷𝐸𝐹”⟩‘𝑖))
125124oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐷) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷))
126120, 125eqeq12d 2753 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐴) = (𝐹 𝐷) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐴) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐷)))
127119oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐵) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵))
128124oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐸) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸))
129127, 128eqeq12d 2753 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐵) = (𝐹 𝐸) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐵) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐸)))
130119oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐶 𝐶) = ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶))
131124oveq1d 7228 . . . . . . 7 ((𝜑𝑖 = 2) → (𝐹 𝐹) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹))
132130, 131eqeq12d 2753 . . . . . 6 ((𝜑𝑖 = 2) → ((𝐶 𝐶) = (𝐹 𝐹) ↔ ((⟨“𝐴𝐵𝐶”⟩‘𝑖) 𝐶) = ((⟨“𝐷𝐸𝐹”⟩‘𝑖) 𝐹)))
133126, 129, 1323anbi123d 1438 . . . . 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 285 . . . 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 4697 . . 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 1485 . . . 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 2737 . . . . . . . 8 (Itv‘𝐺) = (Itv‘𝐺)
13816, 17, 137, 19, 1, 26tgcgrtriv 26575 . . . . . . 7 (𝜑 → (𝐴 𝐴) = (𝐷 𝐷))
13916, 17, 137, 19, 2, 27tgcgrtriv 26575 . . . . . . 7 (𝜑 → (𝐵 𝐵) = (𝐸 𝐸))
14016, 17, 137, 19, 3, 28tgcgrtriv 26575 . . . . . . 7 (𝜑 → (𝐶 𝐶) = (𝐹 𝐹))
141138, 139, 1403jca 1130 . . . . . 6 (𝜑 → ((𝐴 𝐴) = (𝐷 𝐷) ∧ (𝐵 𝐵) = (𝐸 𝐸) ∧ (𝐶 𝐶) = (𝐹 𝐹)))
142141biantrurd 536 . . . . 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 771 . . . . . . 7 ((𝜑 ∧ ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷))) → (𝐴 𝐵) = (𝐷 𝐸))
144 simpr 488 . . . . . . . 8 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → (𝐴 𝐵) = (𝐷 𝐸))
14519adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐺 ∈ TarskiG)
1461adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐴𝑃)
1472adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐵𝑃)
14826adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐷𝑃)
14927adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → 𝐸𝑃)
15016, 17, 137, 145, 146, 147, 148, 149, 144tgcgrcomlr 26571 . . . . . . . 8 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → (𝐵 𝐴) = (𝐸 𝐷))
151144, 150jca 515 . . . . . . 7 ((𝜑 ∧ (𝐴 𝐵) = (𝐷 𝐸)) → ((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)))
152143, 151impbida 801 . . . . . 6 (𝜑 → (((𝐴 𝐵) = (𝐷 𝐸) ∧ (𝐵 𝐴) = (𝐸 𝐷)) ↔ (𝐴 𝐵) = (𝐷 𝐸)))
153 simprl 771 . . . . . . 7 ((𝜑 ∧ ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸))) → (𝐵 𝐶) = (𝐸 𝐹))
154 simpr 488 . . . . . . . 8 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → (𝐵 𝐶) = (𝐸 𝐹))
15519adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐺 ∈ TarskiG)
1562adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐵𝑃)
1573adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐶𝑃)
15827adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐸𝑃)
15928adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → 𝐹𝑃)
16016, 17, 137, 155, 156, 157, 158, 159, 154tgcgrcomlr 26571 . . . . . . . 8 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → (𝐶 𝐵) = (𝐹 𝐸))
161154, 160jca 515 . . . . . . 7 ((𝜑 ∧ (𝐵 𝐶) = (𝐸 𝐹)) → ((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)))
162153, 161impbida 801 . . . . . 6 (𝜑 → (((𝐵 𝐶) = (𝐸 𝐹) ∧ (𝐶 𝐵) = (𝐹 𝐸)) ↔ (𝐵 𝐶) = (𝐸 𝐹)))
163 simprr 773 . . . . . . 7 ((𝜑 ∧ ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷))) → (𝐶 𝐴) = (𝐹 𝐷))
16419adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐺 ∈ TarskiG)
1653adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐶𝑃)
1661adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐴𝑃)
16728adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐹𝑃)
16826adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → 𝐷𝑃)
169 simpr 488 . . . . . . . . 9 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → (𝐶 𝐴) = (𝐹 𝐷))
17016, 17, 137, 164, 165, 166, 167, 168, 169tgcgrcomlr 26571 . . . . . . . 8 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → (𝐴 𝐶) = (𝐷 𝐹))
171170, 169jca 515 . . . . . . 7 ((𝜑 ∧ (𝐶 𝐴) = (𝐹 𝐷)) → ((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)))
172163, 171impbida 801 . . . . . 6 (𝜑 → (((𝐴 𝐶) = (𝐷 𝐹) ∧ (𝐶 𝐴) = (𝐹 𝐷)) ↔ (𝐶 𝐴) = (𝐹 𝐷)))
173152, 162, 1723anbi123d 1438 . . . . 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 284 . . . 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, 174syl5bb 286 . . 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 283 . 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 314 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wss 3866  {ctp 4545   class class class wbr 5053  dom cdm 5551  wf 6376  cfv 6380  (class class class)co 7213  cr 10728  0cc0 10729  1c1 10730  2c2 11885  3c3 11886  ..^cfzo 13238  chash 13896  Word cword 14069  ⟨“cs3 14407  Basecbs 16760  distcds 16811  TarskiGcstrkg 26521  Itvcitv 26527  cgrGccgrg 26601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-fzo 13239  df-hash 13897  df-word 14070  df-concat 14126  df-s1 14153  df-s2 14413  df-s3 14414  df-trkgc 26539  df-trkgcb 26541  df-trkg 26544  df-cgrg 26602
This theorem is referenced by:  trgcgr  26607  cgr3simp1  26611  cgr3simp2  26612  cgr3simp3  26613  dfcgrg2  26954
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