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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 2737 | . . . . . . . . 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 28723 | . . . . . . . . . 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 28652 | . . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑥 = 𝑋) | 
| 18 | eqidd 2737 | . . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑣 = 𝑣) | |
| 19 | 1, 17, 18 | s3eqd 14904 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 〈“𝑢𝑥𝑣”〉 = 〈“𝑢𝑋𝑣”〉) | 
| 20 | 19 | eleq1d 2825 | . . . . . . 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 3166 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) → (∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) → ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) | 
| 23 | 22 | ralimdva 3166 | . . . 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 28721 | . . . 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 3161 | . 2 ⊢ ((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) | 
| 28 | s3eq2 14910 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 〈“𝑢𝑥𝑣”〉 = 〈“𝑢𝑋𝑣”〉) | |
| 29 | 28 | eleq1d 2825 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) ↔ 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) | 
| 30 | 29 | 2ralbidv 3220 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) | 
| 31 | 30 | rspcev 3621 | . . . 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 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ∩ cin 3949 class class class wbr 5142 ran crn 5685 ‘cfv 6560 〈“cs3 14882 Basecbs 17248 distcds 17307 TarskiGcstrkg 28436 Itvcitv 28442 LineGclng 28443 ∟Gcrag 28702 ⟂Gcperpg 28704 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-hash 14371 df-word 14554 df-concat 14610 df-s1 14635 df-s2 14888 df-s3 14889 df-trkgc 28457 df-trkgb 28458 df-trkgcb 28459 df-trkg 28462 df-cgrg 28520 df-mir 28662 df-rag 28703 df-perpg 28705 | 
| This theorem is referenced by: isperp2d 28725 ragperp 28726 foot 28731 | 
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