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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 2738 | . . . . . . . . 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 784 | . . . . . . . . . . 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 28722 | . . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝐴 ≠ 𝐵) | 
| 14 | simpllr 776 | . . . . . . . . . 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 28651 | . . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑥 = 𝑋) | 
| 18 | eqidd 2738 | . . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑣 = 𝑣) | |
| 19 | 1, 17, 18 | s3eqd 14903 | . . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 〈“𝑢𝑥𝑣”〉 = 〈“𝑢𝑋𝑣”〉) | 
| 20 | 19 | eleq1d 2826 | . . . . . . 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 3167 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) ∧ 𝑥 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐴) → (∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) → ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) | 
| 23 | 22 | ralimdva 3167 | . . . 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 28720 | . . . 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 3162 | . 2 ⊢ ((𝜑 ∧ 𝐴(⟂G‘𝐺)𝐵) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) | 
| 28 | s3eq2 14909 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 〈“𝑢𝑥𝑣”〉 = 〈“𝑢𝑋𝑣”〉) | |
| 29 | 28 | eleq1d 2826 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) ↔ 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) | 
| 30 | 29 | 2ralbidv 3221 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝐺) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) | 
| 31 | 30 | rspcev 3622 | . . . 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 801 | 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 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 class class class wbr 5143 ran crn 5686 ‘cfv 6561 〈“cs3 14881 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 ∟Gcrag 28701 ⟂Gcperpg 28703 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 df-cgrg 28519 df-mir 28661 df-rag 28702 df-perpg 28704 | 
| This theorem is referenced by: isperp2d 28724 ragperp 28725 foot 28730 | 
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