Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isperp2d | Structured version Visualization version GIF version |
Description: One direction of isperp2 26500. (Contributed by Thierry Arnoux, 10-Nov-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)) |
isperp2d.u | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
isperp2d.v | ⊢ (𝜑 → 𝑉 ∈ 𝐵) |
isperp2d.p | ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵) |
Ref | Expression |
---|---|
isperp2d | ⊢ (𝜑 → 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isperp2d.p | . . 3 ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵) | |
2 | isperp.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
3 | isperp.d | . . . 4 ⊢ − = (dist‘𝐺) | |
4 | isperp.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | isperp.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | isperp.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | isperp.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) | |
8 | isperp2.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) | |
9 | isperp2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | isperp2 26500 | . . 3 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
11 | 1, 10 | mpbid 234 | . 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) |
12 | isperp2d.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
13 | isperp2d.v | . . 3 ⊢ (𝜑 → 𝑉 ∈ 𝐵) | |
14 | id 22 | . . . . . 6 ⊢ (𝑢 = 𝑈 → 𝑢 = 𝑈) | |
15 | eqidd 2822 | . . . . . 6 ⊢ (𝑢 = 𝑈 → 𝑋 = 𝑋) | |
16 | eqidd 2822 | . . . . . 6 ⊢ (𝑢 = 𝑈 → 𝑣 = 𝑣) | |
17 | 14, 15, 16 | s3eqd 14225 | . . . . 5 ⊢ (𝑢 = 𝑈 → 〈“𝑢𝑋𝑣”〉 = 〈“𝑈𝑋𝑣”〉) |
18 | 17 | eleq1d 2897 | . . . 4 ⊢ (𝑢 = 𝑈 → (〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺) ↔ 〈“𝑈𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
19 | eqidd 2822 | . . . . . 6 ⊢ (𝑣 = 𝑉 → 𝑈 = 𝑈) | |
20 | eqidd 2822 | . . . . . 6 ⊢ (𝑣 = 𝑉 → 𝑋 = 𝑋) | |
21 | id 22 | . . . . . 6 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉) | |
22 | 19, 20, 21 | s3eqd 14225 | . . . . 5 ⊢ (𝑣 = 𝑉 → 〈“𝑈𝑋𝑣”〉 = 〈“𝑈𝑋𝑉”〉) |
23 | 22 | eleq1d 2897 | . . . 4 ⊢ (𝑣 = 𝑉 → (〈“𝑈𝑋𝑣”〉 ∈ (∟G‘𝐺) ↔ 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺))) |
24 | 18, 23 | rspc2v 3632 | . . 3 ⊢ ((𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐵) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺) → 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺))) |
25 | 12, 13, 24 | syl2anc 586 | . 2 ⊢ (𝜑 → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺) → 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺))) |
26 | 11, 25 | mpd 15 | 1 ⊢ (𝜑 → 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∩ cin 3934 class class class wbr 5065 ran crn 5555 ‘cfv 6354 〈“cs3 14203 Basecbs 16482 distcds 16573 TarskiGcstrkg 26215 Itvcitv 26221 LineGclng 26222 ∟Gcrag 26478 ⟂Gcperpg 26480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-s2 14209 df-s3 14210 df-trkgc 26233 df-trkgb 26234 df-trkgcb 26235 df-trkg 26238 df-cgrg 26296 df-mir 26438 df-rag 26479 df-perpg 26481 |
This theorem is referenced by: perprag 26511 |
Copyright terms: Public domain | W3C validator |