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| Mirrors > Home > MPE Home > Th. List > isperp2 | Structured version Visualization version GIF version | ||
| Description: Property for 2 lines A, B, intersecting at a point X to be perpendicular. Item (i) of definition 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-Oct-2019.) |
| Ref | Expression |
|---|---|
| isperp.p | ⊢ 𝑃 = (Base‘𝐺) |
| isperp.d | ⊢ − = (dist‘𝐺) |
| isperp.i | ⊢ 𝐼 = (Itv‘𝐺) |
| isperp.l | ⊢ 𝐿 = (LineG‘𝐺) |
| isperp.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| isperp.a | ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) |
| isperp2.b | ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) |
| isperp2.x | ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
| Ref | Expression |
|---|---|
| isperp2 | ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqidd 2734 | . . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑢 = 𝑢) | |
| 2 | isperp.p | . . . . . . . . . 10 ⊢ 𝑃 = (Base‘𝐺) | |
| 3 | isperp.i | . . . . . . . . . 10 ⊢ 𝐼 = (Itv‘𝐺) | |
| 4 | isperp.l | . . . . . . . . . 10 ⊢ 𝐿 = (LineG‘𝐺) | |
| 5 | isperp.g | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 6 | 5 | ad4antr 732 | . . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝐺 ∈ TarskiG) |
| 7 | isperp.a | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) | |
| 8 | 7 | ad4antr 732 | . . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝐴 ∈ ran 𝐿) |
| 9 | isperp2.b | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) | |
| 10 | 9 | ad4antr 732 | . . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝐵 ∈ ran 𝐿) |
| 11 | isperp.d | . . . . . . . . . . 11 ⊢ − = (dist‘𝐺) | |
| 12 | simp-4r 783 | . . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝐴(⟂G‘𝐺)𝐵) | |
| 13 | 2, 11, 3, 4, 6, 8, 10, 12 | perpneq 28702 | . . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝐴 ≠ 𝐵) |
| 14 | simpllr 775 | . . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑥 ∈ (𝐴 ∩ 𝐵)) | |
| 15 | isperp2.x | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) | |
| 16 | 15 | ad4antr 732 | . . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
| 17 | 2, 3, 4, 6, 8, 10, 13, 14, 16 | tglineineq 28631 | . . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑥 = 𝑋) |
| 18 | eqidd 2734 | . . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑣 = 𝑣) | |
| 19 | 1, 17, 18 | s3eqd 14781 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 〈“𝑢𝑥𝑣”〉 = 〈“𝑢𝑋𝑣”〉) |
| 20 | 19 | eleq1d 2818 | . . . . . . 7 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → (〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) ↔ 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
| 21 | 20 | biimpd 229 | . . . . . 6 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → (〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) → 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
| 22 | 21 | ralimdva 3146 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) → (∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) → ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
| 23 | 22 | ralimdva 3146 | . . . 4 ⊢ (((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
| 24 | 23 | imp 406 | . . 3 ⊢ ((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺)) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) |
| 25 | 2, 11, 3, 4, 5, 7, 9 | isperp 28700 | . . . 4 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |
| 26 | 25 | biimpa 476 | . . 3 ⊢ ((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) → ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺)) |
| 27 | 24, 26 | r19.29a 3142 | . 2 ⊢ ((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) |
| 28 | s3eq2 14787 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 〈“𝑢𝑥𝑣”〉 = 〈“𝑢𝑋𝑣”〉) | |
| 29 | 28 | eleq1d 2818 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) ↔ 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
| 30 | 29 | 2ralbidv 3198 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
| 31 | 30 | rspcev 3574 | . . . 4 ⊢ ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) → ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺)) |
| 32 | 15, 31 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) → ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺)) |
| 33 | 25 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) → (𝐴(⟂G‘𝐺)𝐵 ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺))) |
| 34 | 32, 33 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) → 𝐴(⟂G‘𝐺)𝐵) |
| 35 | 27, 34 | impbida 800 | 1 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 ∩ cin 3898 class class class wbr 5095 ran crn 5622 ‘cfv 6489 〈“cs3 14759 Basecbs 17130 distcds 17180 TarskiGcstrkg 28415 Itvcitv 28421 LineGclng 28422 ∟Gcrag 28681 ⟂Gcperpg 28683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-oadd 8398 df-er 8631 df-map 8761 df-pm 8762 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-dju 9804 df-card 9842 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-nn 12136 df-2 12198 df-3 12199 df-n0 12392 df-xnn0 12465 df-z 12479 df-uz 12743 df-fz 13418 df-fzo 13565 df-hash 14248 df-word 14431 df-concat 14488 df-s1 14514 df-s2 14765 df-s3 14766 df-trkgc 28436 df-trkgb 28437 df-trkgcb 28438 df-trkg 28441 df-cgrg 28499 df-mir 28641 df-rag 28682 df-perpg 28684 |
| This theorem is referenced by: isperp2d 28704 ragperp 28705 foot 28710 |
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