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Theorem brafs 33679
Description: Binary relation form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.)
Hypotheses
Ref Expression
brafs.p 𝑃 = (Baseβ€˜πΊ)
brafs.d βˆ’ = (distβ€˜πΊ)
brafs.i 𝐼 = (Itvβ€˜πΊ)
brafs.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
brafs.o 𝑂 = (AFSβ€˜πΊ)
brafs.1 (πœ‘ β†’ 𝐴 ∈ 𝑃)
brafs.2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
brafs.3 (πœ‘ β†’ 𝐢 ∈ 𝑃)
brafs.4 (πœ‘ β†’ 𝐷 ∈ 𝑃)
brafs.5 (πœ‘ β†’ 𝑋 ∈ 𝑃)
brafs.6 (πœ‘ β†’ π‘Œ ∈ 𝑃)
brafs.7 (πœ‘ β†’ 𝑍 ∈ 𝑃)
brafs.8 (πœ‘ β†’ π‘Š ∈ 𝑃)
Assertion
Ref Expression
brafs (πœ‘ β†’ (⟨⟨𝐴, 𝐡⟩, ⟨𝐢, π·βŸ©βŸ©π‘‚βŸ¨βŸ¨π‘‹, π‘ŒβŸ©, βŸ¨π‘, π‘ŠβŸ©βŸ© ↔ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ π‘Œ ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ) ∧ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑍)) ∧ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ π‘Š) ∧ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ π‘Š)))))

Proof of Theorem brafs
Dummy variables π‘Ž 𝑏 𝑐 𝑑 𝑒 𝑓 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7415 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘ŽπΌπ‘) = (𝐴𝐼𝑐))
21eleq2d 2819 . . . 4 (π‘Ž = 𝐴 β†’ (𝑏 ∈ (π‘ŽπΌπ‘) ↔ 𝑏 ∈ (𝐴𝐼𝑐)))
32anbi1d 630 . . 3 (π‘Ž = 𝐴 β†’ ((𝑏 ∈ (π‘ŽπΌπ‘) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ↔ (𝑏 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (π‘₯𝐼𝑧))))
4 oveq1 7415 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘Ž βˆ’ 𝑏) = (𝐴 βˆ’ 𝑏))
54eqeq1d 2734 . . . 4 (π‘Ž = 𝐴 β†’ ((π‘Ž βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦) ↔ (𝐴 βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦)))
65anbi1d 630 . . 3 (π‘Ž = 𝐴 β†’ (((π‘Ž βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦) ∧ (𝑏 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)) ↔ ((𝐴 βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦) ∧ (𝑏 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧))))
7 oveq1 7415 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘Ž βˆ’ 𝑑) = (𝐴 βˆ’ 𝑑))
87eqeq1d 2734 . . . 4 (π‘Ž = 𝐴 β†’ ((π‘Ž βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ↔ (𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀)))
98anbi1d 630 . . 3 (π‘Ž = 𝐴 β†’ (((π‘Ž βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝑏 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀)) ↔ ((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝑏 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀))))
103, 6, 93anbi123d 1436 . 2 (π‘Ž = 𝐴 β†’ (((𝑏 ∈ (π‘ŽπΌπ‘) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((π‘Ž βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦) ∧ (𝑏 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)) ∧ ((π‘Ž βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝑏 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀))) ↔ ((𝑏 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦) ∧ (𝑏 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝑏 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀)))))
11 eleq1 2821 . . . 4 (𝑏 = 𝐡 β†’ (𝑏 ∈ (𝐴𝐼𝑐) ↔ 𝐡 ∈ (𝐴𝐼𝑐)))
1211anbi1d 630 . . 3 (𝑏 = 𝐡 β†’ ((𝑏 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ↔ (𝐡 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (π‘₯𝐼𝑧))))
13 oveq2 7416 . . . . 5 (𝑏 = 𝐡 β†’ (𝐴 βˆ’ 𝑏) = (𝐴 βˆ’ 𝐡))
1413eqeq1d 2734 . . . 4 (𝑏 = 𝐡 β†’ ((𝐴 βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦) ↔ (𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦)))
15 oveq1 7415 . . . . 5 (𝑏 = 𝐡 β†’ (𝑏 βˆ’ 𝑐) = (𝐡 βˆ’ 𝑐))
1615eqeq1d 2734 . . . 4 (𝑏 = 𝐡 β†’ ((𝑏 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧) ↔ (𝐡 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)))
1714, 16anbi12d 631 . . 3 (𝑏 = 𝐡 β†’ (((𝐴 βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦) ∧ (𝑏 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)) ↔ ((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧))))
18 oveq1 7415 . . . . 5 (𝑏 = 𝐡 β†’ (𝑏 βˆ’ 𝑑) = (𝐡 βˆ’ 𝑑))
1918eqeq1d 2734 . . . 4 (𝑏 = 𝐡 β†’ ((𝑏 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀) ↔ (𝐡 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀)))
2019anbi2d 629 . . 3 (𝑏 = 𝐡 β†’ (((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝑏 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀)) ↔ ((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀))))
2112, 17, 203anbi123d 1436 . 2 (𝑏 = 𝐡 β†’ (((𝑏 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦) ∧ (𝑏 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝑏 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀))) ↔ ((𝐡 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀)))))
22 oveq2 7416 . . . . 5 (𝑐 = 𝐢 β†’ (𝐴𝐼𝑐) = (𝐴𝐼𝐢))
2322eleq2d 2819 . . . 4 (𝑐 = 𝐢 β†’ (𝐡 ∈ (𝐴𝐼𝑐) ↔ 𝐡 ∈ (𝐴𝐼𝐢)))
2423anbi1d 630 . . 3 (𝑐 = 𝐢 β†’ ((𝐡 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ↔ (𝐡 ∈ (𝐴𝐼𝐢) ∧ 𝑦 ∈ (π‘₯𝐼𝑧))))
25 oveq2 7416 . . . . 5 (𝑐 = 𝐢 β†’ (𝐡 βˆ’ 𝑐) = (𝐡 βˆ’ 𝐢))
2625eqeq1d 2734 . . . 4 (𝑐 = 𝐢 β†’ ((𝐡 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧) ↔ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧)))
2726anbi2d 629 . . 3 (𝑐 = 𝐢 β†’ (((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)) ↔ ((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧))))
2824, 273anbi12d 1437 . 2 (𝑐 = 𝐢 β†’ (((𝐡 ∈ (𝐴𝐼𝑐) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀))) ↔ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀)))))
29 oveq2 7416 . . . . 5 (𝑑 = 𝐷 β†’ (𝐴 βˆ’ 𝑑) = (𝐴 βˆ’ 𝐷))
3029eqeq1d 2734 . . . 4 (𝑑 = 𝐷 β†’ ((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ↔ (𝐴 βˆ’ 𝐷) = (π‘₯ βˆ’ 𝑀)))
31 oveq2 7416 . . . . 5 (𝑑 = 𝐷 β†’ (𝐡 βˆ’ 𝑑) = (𝐡 βˆ’ 𝐷))
3231eqeq1d 2734 . . . 4 (𝑑 = 𝐷 β†’ ((𝐡 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀) ↔ (𝐡 βˆ’ 𝐷) = (𝑦 βˆ’ 𝑀)))
3330, 32anbi12d 631 . . 3 (𝑑 = 𝐷 β†’ (((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀)) ↔ ((𝐴 βˆ’ 𝐷) = (π‘₯ βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (𝑦 βˆ’ 𝑀))))
34333anbi3d 1442 . 2 (𝑑 = 𝐷 β†’ (((𝐡 ∈ (𝐴𝐼𝐢) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀))) ↔ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝐷) = (π‘₯ βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (𝑦 βˆ’ 𝑀)))))
35 oveq1 7415 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯𝐼𝑧) = (𝑋𝐼𝑧))
3635eleq2d 2819 . . . 4 (π‘₯ = 𝑋 β†’ (𝑦 ∈ (π‘₯𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
3736anbi2d 629 . . 3 (π‘₯ = 𝑋 β†’ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ↔ (𝐡 ∈ (𝐴𝐼𝐢) ∧ 𝑦 ∈ (𝑋𝐼𝑧))))
38 oveq1 7415 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯ βˆ’ 𝑦) = (𝑋 βˆ’ 𝑦))
3938eqeq2d 2743 . . . 4 (π‘₯ = 𝑋 β†’ ((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ↔ (𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ 𝑦)))
4039anbi1d 630 . . 3 (π‘₯ = 𝑋 β†’ (((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧)) ↔ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧))))
41 oveq1 7415 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯ βˆ’ 𝑀) = (𝑋 βˆ’ 𝑀))
4241eqeq2d 2743 . . . 4 (π‘₯ = 𝑋 β†’ ((𝐴 βˆ’ 𝐷) = (π‘₯ βˆ’ 𝑀) ↔ (𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀)))
4342anbi1d 630 . . 3 (π‘₯ = 𝑋 β†’ (((𝐴 βˆ’ 𝐷) = (π‘₯ βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (𝑦 βˆ’ 𝑀)) ↔ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (𝑦 βˆ’ 𝑀))))
4437, 40, 433anbi123d 1436 . 2 (π‘₯ = 𝑋 β†’ (((𝐡 ∈ (𝐴𝐼𝐢) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝐡) = (π‘₯ βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝐷) = (π‘₯ βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (𝑦 βˆ’ 𝑀))) ↔ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ 𝑦 ∈ (𝑋𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (𝑦 βˆ’ 𝑀)))))
45 eleq1 2821 . . . 4 (𝑦 = π‘Œ β†’ (𝑦 ∈ (𝑋𝐼𝑧) ↔ π‘Œ ∈ (𝑋𝐼𝑧)))
4645anbi2d 629 . . 3 (𝑦 = π‘Œ β†’ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ (𝐡 ∈ (𝐴𝐼𝐢) ∧ π‘Œ ∈ (𝑋𝐼𝑧))))
47 oveq2 7416 . . . . 5 (𝑦 = π‘Œ β†’ (𝑋 βˆ’ 𝑦) = (𝑋 βˆ’ π‘Œ))
4847eqeq2d 2743 . . . 4 (𝑦 = π‘Œ β†’ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ 𝑦) ↔ (𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ)))
49 oveq1 7415 . . . . 5 (𝑦 = π‘Œ β†’ (𝑦 βˆ’ 𝑧) = (π‘Œ βˆ’ 𝑧))
5049eqeq2d 2743 . . . 4 (𝑦 = π‘Œ β†’ ((𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧) ↔ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑧)))
5148, 50anbi12d 631 . . 3 (𝑦 = π‘Œ β†’ (((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧)) ↔ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ) ∧ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑧))))
52 oveq1 7415 . . . . 5 (𝑦 = π‘Œ β†’ (𝑦 βˆ’ 𝑀) = (π‘Œ βˆ’ 𝑀))
5352eqeq2d 2743 . . . 4 (𝑦 = π‘Œ β†’ ((𝐡 βˆ’ 𝐷) = (𝑦 βˆ’ 𝑀) ↔ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ 𝑀)))
5453anbi2d 629 . . 3 (𝑦 = π‘Œ β†’ (((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (𝑦 βˆ’ 𝑀)) ↔ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ 𝑀))))
5546, 51, 543anbi123d 1436 . 2 (𝑦 = π‘Œ β†’ (((𝐡 ∈ (𝐴𝐼𝐢) ∧ 𝑦 ∈ (𝑋𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ 𝑦) ∧ (𝐡 βˆ’ 𝐢) = (𝑦 βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (𝑦 βˆ’ 𝑀))) ↔ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ π‘Œ ∈ (𝑋𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ) ∧ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ 𝑀)))))
56 oveq2 7416 . . . . 5 (𝑧 = 𝑍 β†’ (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
5756eleq2d 2819 . . . 4 (𝑧 = 𝑍 β†’ (π‘Œ ∈ (𝑋𝐼𝑧) ↔ π‘Œ ∈ (𝑋𝐼𝑍)))
5857anbi2d 629 . . 3 (𝑧 = 𝑍 β†’ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ π‘Œ ∈ (𝑋𝐼𝑧)) ↔ (𝐡 ∈ (𝐴𝐼𝐢) ∧ π‘Œ ∈ (𝑋𝐼𝑍))))
59 oveq2 7416 . . . . 5 (𝑧 = 𝑍 β†’ (π‘Œ βˆ’ 𝑧) = (π‘Œ βˆ’ 𝑍))
6059eqeq2d 2743 . . . 4 (𝑧 = 𝑍 β†’ ((𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑧) ↔ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑍)))
6160anbi2d 629 . . 3 (𝑧 = 𝑍 β†’ (((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ) ∧ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑧)) ↔ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ) ∧ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑍))))
6258, 613anbi12d 1437 . 2 (𝑧 = 𝑍 β†’ (((𝐡 ∈ (𝐴𝐼𝐢) ∧ π‘Œ ∈ (𝑋𝐼𝑧)) ∧ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ) ∧ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑧)) ∧ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ 𝑀))) ↔ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ π‘Œ ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ) ∧ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑍)) ∧ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ 𝑀)))))
63 oveq2 7416 . . . . 5 (𝑀 = π‘Š β†’ (𝑋 βˆ’ 𝑀) = (𝑋 βˆ’ π‘Š))
6463eqeq2d 2743 . . . 4 (𝑀 = π‘Š β†’ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ↔ (𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ π‘Š)))
65 oveq2 7416 . . . . 5 (𝑀 = π‘Š β†’ (π‘Œ βˆ’ 𝑀) = (π‘Œ βˆ’ π‘Š))
6665eqeq2d 2743 . . . 4 (𝑀 = π‘Š β†’ ((𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ 𝑀) ↔ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ π‘Š)))
6764, 66anbi12d 631 . . 3 (𝑀 = π‘Š β†’ (((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ 𝑀)) ↔ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ π‘Š) ∧ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ π‘Š))))
68673anbi3d 1442 . 2 (𝑀 = π‘Š β†’ (((𝐡 ∈ (𝐴𝐼𝐢) ∧ π‘Œ ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ) ∧ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑍)) ∧ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ 𝑀) ∧ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ 𝑀))) ↔ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ π‘Œ ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ) ∧ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑍)) ∧ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ π‘Š) ∧ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ π‘Š)))))
69 brafs.o . . 3 𝑂 = (AFSβ€˜πΊ)
70 brafs.p . . . 4 𝑃 = (Baseβ€˜πΊ)
71 brafs.d . . . 4 βˆ’ = (distβ€˜πΊ)
72 brafs.i . . . 4 𝐼 = (Itvβ€˜πΊ)
73 brafs.g . . . 4 (πœ‘ β†’ 𝐺 ∈ TarskiG)
7470, 71, 72, 73afsval 33678 . . 3 (πœ‘ β†’ (AFSβ€˜πΊ) = {βŸ¨π‘’, π‘“βŸ© ∣ βˆƒπ‘Ž ∈ 𝑃 βˆƒπ‘ ∈ 𝑃 βˆƒπ‘ ∈ 𝑃 βˆƒπ‘‘ ∈ 𝑃 βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 βˆƒπ‘€ ∈ 𝑃 (𝑒 = βŸ¨βŸ¨π‘Ž, π‘βŸ©, βŸ¨π‘, π‘‘βŸ©βŸ© ∧ 𝑓 = ⟨⟨π‘₯, π‘¦βŸ©, βŸ¨π‘§, π‘€βŸ©βŸ© ∧ ((𝑏 ∈ (π‘ŽπΌπ‘) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((π‘Ž βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦) ∧ (𝑏 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)) ∧ ((π‘Ž βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝑏 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀))))})
7569, 74eqtrid 2784 . 2 (πœ‘ β†’ 𝑂 = {βŸ¨π‘’, π‘“βŸ© ∣ βˆƒπ‘Ž ∈ 𝑃 βˆƒπ‘ ∈ 𝑃 βˆƒπ‘ ∈ 𝑃 βˆƒπ‘‘ ∈ 𝑃 βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 βˆƒπ‘€ ∈ 𝑃 (𝑒 = βŸ¨βŸ¨π‘Ž, π‘βŸ©, βŸ¨π‘, π‘‘βŸ©βŸ© ∧ 𝑓 = ⟨⟨π‘₯, π‘¦βŸ©, βŸ¨π‘§, π‘€βŸ©βŸ© ∧ ((𝑏 ∈ (π‘ŽπΌπ‘) ∧ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ ((π‘Ž βˆ’ 𝑏) = (π‘₯ βˆ’ 𝑦) ∧ (𝑏 βˆ’ 𝑐) = (𝑦 βˆ’ 𝑧)) ∧ ((π‘Ž βˆ’ 𝑑) = (π‘₯ βˆ’ 𝑀) ∧ (𝑏 βˆ’ 𝑑) = (𝑦 βˆ’ 𝑀))))})
76 brafs.1 . 2 (πœ‘ β†’ 𝐴 ∈ 𝑃)
77 brafs.2 . 2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
78 brafs.3 . 2 (πœ‘ β†’ 𝐢 ∈ 𝑃)
79 brafs.4 . 2 (πœ‘ β†’ 𝐷 ∈ 𝑃)
80 brafs.5 . 2 (πœ‘ β†’ 𝑋 ∈ 𝑃)
81 brafs.6 . 2 (πœ‘ β†’ π‘Œ ∈ 𝑃)
82 brafs.7 . 2 (πœ‘ β†’ 𝑍 ∈ 𝑃)
83 brafs.8 . 2 (πœ‘ β†’ π‘Š ∈ 𝑃)
8410, 21, 28, 34, 44, 55, 62, 68, 75, 76, 77, 78, 79, 80, 81, 82, 83br8d 31834 1 (πœ‘ β†’ (⟨⟨𝐴, 𝐡⟩, ⟨𝐢, π·βŸ©βŸ©π‘‚βŸ¨βŸ¨π‘‹, π‘ŒβŸ©, βŸ¨π‘, π‘ŠβŸ©βŸ© ↔ ((𝐡 ∈ (𝐴𝐼𝐢) ∧ π‘Œ ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 βˆ’ 𝐡) = (𝑋 βˆ’ π‘Œ) ∧ (𝐡 βˆ’ 𝐢) = (π‘Œ βˆ’ 𝑍)) ∧ ((𝐴 βˆ’ 𝐷) = (𝑋 βˆ’ π‘Š) ∧ (𝐡 βˆ’ 𝐷) = (π‘Œ βˆ’ π‘Š)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  βŸ¨cop 4634   class class class wbr 5148  {copab 5210  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  distcds 17205  TarskiGcstrkg 27675  Itvcitv 27681  AFScafs 33676
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-afs 33677
This theorem is referenced by:  tg5segofs  33680
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