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Theorem axtgupdim2ALTV 33680
Description: Alternate version of axtgupdim2 27722. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.)
Hypotheses
Ref Expression
istrkg2d.p 𝑃 = (Baseβ€˜πΊ)
istrkg2d.d βˆ’ = (distβ€˜πΊ)
istrkg2d.i 𝐼 = (Itvβ€˜πΊ)
axtgupdim2ALTV.x (πœ‘ β†’ 𝑋 ∈ 𝑃)
axtgupdim2ALTV.y (πœ‘ β†’ π‘Œ ∈ 𝑃)
axtgupdim2ALTV.z (πœ‘ β†’ 𝑍 ∈ 𝑃)
axtgupdim2ALTV.u (πœ‘ β†’ π‘ˆ ∈ 𝑃)
axtgupdim2ALTV.v (πœ‘ β†’ 𝑉 ∈ 𝑃)
axtgupdim2ALTV.0 (πœ‘ β†’ π‘ˆ β‰  𝑉)
axtgupdim2ALTV.1 (πœ‘ β†’ (𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑉))
axtgupdim2ALTV.2 (πœ‘ β†’ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑉))
axtgupdim2ALTV.3 (πœ‘ β†’ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑉))
axtgupdim2ALTV.g (πœ‘ β†’ 𝐺 ∈ TarskiG2D)
Assertion
Ref Expression
axtgupdim2ALTV (πœ‘ β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))

Proof of Theorem axtgupdim2ALTV
Dummy variables 𝑒 𝑣 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axtgupdim2ALTV.1 . . 3 (πœ‘ β†’ (𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑉))
2 axtgupdim2ALTV.2 . . 3 (πœ‘ β†’ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑉))
3 axtgupdim2ALTV.3 . . 3 (πœ‘ β†’ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑉))
41, 2, 33jca 1129 . 2 (πœ‘ β†’ ((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑉) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑉) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑉)))
5 axtgupdim2ALTV.0 . 2 (πœ‘ β†’ π‘ˆ β‰  𝑉)
6 axtgupdim2ALTV.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG2D)
7 istrkg2d.p . . . . . . 7 𝑃 = (Baseβ€˜πΊ)
8 istrkg2d.d . . . . . . 7 βˆ’ = (distβ€˜πΊ)
9 istrkg2d.i . . . . . . 7 𝐼 = (Itvβ€˜πΊ)
107, 8, 9istrkg2d 33678 . . . . . 6 (𝐺 ∈ TarskiG2D ↔ (𝐺 ∈ V ∧ (βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))))))
116, 10sylib 217 . . . . 5 (πœ‘ β†’ (𝐺 ∈ V ∧ (βˆƒπ‘₯ ∈ 𝑃 βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘§ ∈ 𝑃 Β¬ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))))))
1211simprrd 773 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))))
13 axtgupdim2ALTV.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝑃)
14 axtgupdim2ALTV.y . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝑃)
15 axtgupdim2ALTV.z . . . . 5 (πœ‘ β†’ 𝑍 ∈ 𝑃)
16 oveq1 7416 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ (π‘₯ βˆ’ 𝑒) = (𝑋 βˆ’ 𝑒))
17 oveq1 7416 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ (π‘₯ βˆ’ 𝑣) = (𝑋 βˆ’ 𝑣))
1816, 17eqeq12d 2749 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ ((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ↔ (𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣)))
19183anbi1d 1441 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ↔ ((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣))))
2019anbi1d 631 . . . . . . . 8 (π‘₯ = 𝑋 β†’ ((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) ↔ (((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣)))
21 oveq1 7416 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (π‘₯𝐼𝑦) = (𝑋𝐼𝑦))
2221eleq2d 2820 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑦)))
23 eleq1 2822 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (π‘₯ ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑦)))
24 oveq1 7416 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (π‘₯𝐼𝑧) = (𝑋𝐼𝑧))
2524eleq2d 2820 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (𝑦 ∈ (π‘₯𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
2622, 23, 253orbi123d 1436 . . . . . . . 8 (π‘₯ = 𝑋 β†’ ((𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))))
2720, 26imbi12d 345 . . . . . . 7 (π‘₯ = 𝑋 β†’ (((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))) ↔ ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)))))
28272ralbidv 3219 . . . . . 6 (π‘₯ = 𝑋 β†’ (βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))) ↔ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)))))
29 oveq1 7416 . . . . . . . . . . 11 (𝑦 = π‘Œ β†’ (𝑦 βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑒))
30 oveq1 7416 . . . . . . . . . . 11 (𝑦 = π‘Œ β†’ (𝑦 βˆ’ 𝑣) = (π‘Œ βˆ’ 𝑣))
3129, 30eqeq12d 2749 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ ((𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ↔ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣)))
32313anbi2d 1442 . . . . . . . . 9 (𝑦 = π‘Œ β†’ (((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ↔ ((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣))))
3332anbi1d 631 . . . . . . . 8 (𝑦 = π‘Œ β†’ ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) ↔ (((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣)))
34 oveq2 7417 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ (𝑋𝐼𝑦) = (π‘‹πΌπ‘Œ))
3534eleq2d 2820 . . . . . . . . 9 (𝑦 = π‘Œ β†’ (𝑧 ∈ (𝑋𝐼𝑦) ↔ 𝑧 ∈ (π‘‹πΌπ‘Œ)))
36 oveq2 7417 . . . . . . . . . 10 (𝑦 = π‘Œ β†’ (𝑧𝐼𝑦) = (π‘§πΌπ‘Œ))
3736eleq2d 2820 . . . . . . . . 9 (𝑦 = π‘Œ β†’ (𝑋 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (π‘§πΌπ‘Œ)))
38 eleq1 2822 . . . . . . . . 9 (𝑦 = π‘Œ β†’ (𝑦 ∈ (𝑋𝐼𝑧) ↔ π‘Œ ∈ (𝑋𝐼𝑧)))
3935, 37, 383orbi123d 1436 . . . . . . . 8 (𝑦 = π‘Œ β†’ ((𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))))
4033, 39imbi12d 345 . . . . . . 7 (𝑦 = π‘Œ β†’ (((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))) ↔ ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧)))))
41402ralbidv 3219 . . . . . 6 (𝑦 = π‘Œ β†’ (βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))) ↔ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧)))))
42 oveq1 7416 . . . . . . . . . . 11 (𝑧 = 𝑍 β†’ (𝑧 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑒))
43 oveq1 7416 . . . . . . . . . . 11 (𝑧 = 𝑍 β†’ (𝑧 βˆ’ 𝑣) = (𝑍 βˆ’ 𝑣))
4442, 43eqeq12d 2749 . . . . . . . . . 10 (𝑧 = 𝑍 β†’ ((𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣) ↔ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)))
45443anbi3d 1443 . . . . . . . . 9 (𝑧 = 𝑍 β†’ (((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ↔ ((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣))))
4645anbi1d 631 . . . . . . . 8 (𝑧 = 𝑍 β†’ ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) ↔ (((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣)))
47 eleq1 2822 . . . . . . . . 9 (𝑧 = 𝑍 β†’ (𝑧 ∈ (π‘‹πΌπ‘Œ) ↔ 𝑍 ∈ (π‘‹πΌπ‘Œ)))
48 oveq1 7416 . . . . . . . . . 10 (𝑧 = 𝑍 β†’ (π‘§πΌπ‘Œ) = (π‘πΌπ‘Œ))
4948eleq2d 2820 . . . . . . . . 9 (𝑧 = 𝑍 β†’ (𝑋 ∈ (π‘§πΌπ‘Œ) ↔ 𝑋 ∈ (π‘πΌπ‘Œ)))
50 oveq2 7417 . . . . . . . . . 10 (𝑧 = 𝑍 β†’ (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
5150eleq2d 2820 . . . . . . . . 9 (𝑧 = 𝑍 β†’ (π‘Œ ∈ (𝑋𝐼𝑧) ↔ π‘Œ ∈ (𝑋𝐼𝑍)))
5247, 49, 513orbi123d 1436 . . . . . . . 8 (𝑧 = 𝑍 β†’ ((𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧)) ↔ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))))
5346, 52imbi12d 345 . . . . . . 7 (𝑧 = 𝑍 β†’ (((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))) ↔ ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))))
54532ralbidv 3219 . . . . . 6 (𝑧 = 𝑍 β†’ (βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘§πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑧))) ↔ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))))
5528, 41, 54rspc3v 3628 . . . . 5 ((𝑋 ∈ 𝑃 ∧ π‘Œ ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) β†’ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))) β†’ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))))
5613, 14, 15, 55syl3anc 1372 . . . 4 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((π‘₯ βˆ’ 𝑒) = (π‘₯ βˆ’ 𝑣) ∧ (𝑦 βˆ’ 𝑒) = (𝑦 βˆ’ 𝑣) ∧ (𝑧 βˆ’ 𝑒) = (𝑧 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))) β†’ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))))
5712, 56mpd 15 . . 3 (πœ‘ β†’ βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))))
58 axtgupdim2ALTV.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑃)
59 axtgupdim2ALTV.v . . . 4 (πœ‘ β†’ 𝑉 ∈ 𝑃)
60 oveq2 7417 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ (𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ π‘ˆ))
6160eqeq1d 2735 . . . . . . . 8 (𝑒 = π‘ˆ β†’ ((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ↔ (𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑣)))
62 oveq2 7417 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ π‘ˆ))
6362eqeq1d 2735 . . . . . . . 8 (𝑒 = π‘ˆ β†’ ((π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ↔ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑣)))
64 oveq2 7417 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ π‘ˆ))
6564eqeq1d 2735 . . . . . . . 8 (𝑒 = π‘ˆ β†’ ((𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣) ↔ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑣)))
6661, 63, 653anbi123d 1437 . . . . . . 7 (𝑒 = π‘ˆ β†’ (((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ↔ ((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑣))))
67 neeq1 3004 . . . . . . 7 (𝑒 = π‘ˆ β†’ (𝑒 β‰  𝑣 ↔ π‘ˆ β‰  𝑣))
6866, 67anbi12d 632 . . . . . 6 (𝑒 = π‘ˆ β†’ ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) ↔ (((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑣)) ∧ π‘ˆ β‰  𝑣)))
6968imbi1d 342 . . . . 5 (𝑒 = π‘ˆ β†’ (((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))) ↔ ((((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑣)) ∧ π‘ˆ β‰  𝑣) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))))
70 oveq2 7417 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (𝑋 βˆ’ 𝑣) = (𝑋 βˆ’ 𝑉))
7170eqeq2d 2744 . . . . . . . 8 (𝑣 = 𝑉 β†’ ((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑣) ↔ (𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑉)))
72 oveq2 7417 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (π‘Œ βˆ’ 𝑣) = (π‘Œ βˆ’ 𝑉))
7372eqeq2d 2744 . . . . . . . 8 (𝑣 = 𝑉 β†’ ((π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑣) ↔ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑉)))
74 oveq2 7417 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (𝑍 βˆ’ 𝑣) = (𝑍 βˆ’ 𝑉))
7574eqeq2d 2744 . . . . . . . 8 (𝑣 = 𝑉 β†’ ((𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑣) ↔ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑉)))
7671, 73, 753anbi123d 1437 . . . . . . 7 (𝑣 = 𝑉 β†’ (((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑣)) ↔ ((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑉) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑉) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑉))))
77 neeq2 3005 . . . . . . 7 (𝑣 = 𝑉 β†’ (π‘ˆ β‰  𝑣 ↔ π‘ˆ β‰  𝑉))
7876, 77anbi12d 632 . . . . . 6 (𝑣 = 𝑉 β†’ ((((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑣)) ∧ π‘ˆ β‰  𝑣) ↔ (((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑉) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑉) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑉)) ∧ π‘ˆ β‰  𝑉)))
7978imbi1d 342 . . . . 5 (𝑣 = 𝑉 β†’ (((((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑣)) ∧ π‘ˆ β‰  𝑣) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))) ↔ ((((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑉) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑉) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑉)) ∧ π‘ˆ β‰  𝑉) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))))
8069, 79rspc2v 3623 . . . 4 ((π‘ˆ ∈ 𝑃 ∧ 𝑉 ∈ 𝑃) β†’ (βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))) β†’ ((((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑉) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑉) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑉)) ∧ π‘ˆ β‰  𝑉) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))))
8158, 59, 80syl2anc 585 . . 3 (πœ‘ β†’ (βˆ€π‘’ ∈ 𝑃 βˆ€π‘£ ∈ 𝑃 ((((𝑋 βˆ’ 𝑒) = (𝑋 βˆ’ 𝑣) ∧ (π‘Œ βˆ’ 𝑒) = (π‘Œ βˆ’ 𝑣) ∧ (𝑍 βˆ’ 𝑒) = (𝑍 βˆ’ 𝑣)) ∧ 𝑒 β‰  𝑣) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))) β†’ ((((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑉) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑉) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑉)) ∧ π‘ˆ β‰  𝑉) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))))
8257, 81mpd 15 . 2 (πœ‘ β†’ ((((𝑋 βˆ’ π‘ˆ) = (𝑋 βˆ’ 𝑉) ∧ (π‘Œ βˆ’ π‘ˆ) = (π‘Œ βˆ’ 𝑉) ∧ (𝑍 βˆ’ π‘ˆ) = (𝑍 βˆ’ 𝑉)) ∧ π‘ˆ β‰  𝑉) β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))))
834, 5, 82mp2and 698 1 (πœ‘ β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∨ w3o 1087   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  distcds 17206  Itvcitv 27684  TarskiG2Dcstrkg2d 33676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-trkg2d 33677
This theorem is referenced by: (None)
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