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Mirrors > Home > MPE Home > Th. List > Mathboxes > axtglowdim2ALTV | Structured version Visualization version GIF version |
Description: Alternate version of axtglowdim2 26254. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
istrkg2d.p | ⊢ 𝑃 = (Base‘𝐺) |
istrkg2d.d | ⊢ − = (dist‘𝐺) |
istrkg2d.i | ⊢ 𝐼 = (Itv‘𝐺) |
axtglowdim2ALTV.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG2D) |
Ref | Expression |
---|---|
axtglowdim2ALTV | ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axtglowdim2ALTV.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG2D) | |
2 | istrkg2d.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
3 | istrkg2d.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
4 | istrkg2d.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | 2, 3, 4 | istrkg2d 31958 | . . . 4 ⊢ (𝐺 ∈ TarskiG2D ↔ (𝐺 ∈ V ∧ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((((𝑥 − 𝑢) = (𝑥 − 𝑣) ∧ (𝑦 − 𝑢) = (𝑦 − 𝑣) ∧ (𝑧 − 𝑢) = (𝑧 − 𝑣)) ∧ 𝑢 ≠ 𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))) |
6 | 1, 5 | sylib 220 | . . 3 ⊢ (𝜑 → (𝐺 ∈ V ∧ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((((𝑥 − 𝑢) = (𝑥 − 𝑣) ∧ (𝑦 − 𝑢) = (𝑦 − 𝑣) ∧ (𝑧 − 𝑢) = (𝑧 − 𝑣)) ∧ 𝑢 ≠ 𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))) |
7 | 6 | simprd 498 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((((𝑥 − 𝑢) = (𝑥 − 𝑣) ∧ (𝑦 − 𝑢) = (𝑦 − 𝑣) ∧ (𝑧 − 𝑢) = (𝑧 − 𝑣)) ∧ 𝑢 ≠ 𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))) |
8 | 7 | simpld 497 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ w3o 1081 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∀wral 3137 ∃wrex 3138 Vcvv 3491 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 distcds 16569 Itvcitv 26220 TarskiG2Dcstrkg2d 31956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-iota 6307 df-fv 6356 df-ov 7152 df-trkg2d 31957 |
This theorem is referenced by: (None) |
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