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Mirrors > Home > MPE Home > Th. List > isperp2d | Structured version Visualization version GIF version |
Description: One direction of isperp2 27543. (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 27543 | . . 3 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
11 | 1, 10 | mpbid 231 | . 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 2737 | . . . . . 6 ⊢ (𝑢 = 𝑈 → 𝑋 = 𝑋) | |
16 | eqidd 2737 | . . . . . 6 ⊢ (𝑢 = 𝑈 → 𝑣 = 𝑣) | |
17 | 14, 15, 16 | s3eqd 14745 | . . . . 5 ⊢ (𝑢 = 𝑈 → 〈“𝑢𝑋𝑣”〉 = 〈“𝑈𝑋𝑣”〉) |
18 | 17 | eleq1d 2822 | . . . 4 ⊢ (𝑢 = 𝑈 → (〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺) ↔ 〈“𝑈𝑋𝑣”〉 ∈ (∟G‘𝐺))) |
19 | eqidd 2737 | . . . . . 6 ⊢ (𝑣 = 𝑉 → 𝑈 = 𝑈) | |
20 | eqidd 2737 | . . . . . 6 ⊢ (𝑣 = 𝑉 → 𝑋 = 𝑋) | |
21 | id 22 | . . . . . 6 ⊢ (𝑣 = 𝑉 → 𝑣 = 𝑉) | |
22 | 19, 20, 21 | s3eqd 14745 | . . . . 5 ⊢ (𝑣 = 𝑉 → 〈“𝑈𝑋𝑣”〉 = 〈“𝑈𝑋𝑉”〉) |
23 | 22 | eleq1d 2822 | . . . 4 ⊢ (𝑣 = 𝑉 → (〈“𝑈𝑋𝑣”〉 ∈ (∟G‘𝐺) ↔ 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺))) |
24 | 18, 23 | rspc2v 3588 | . . 3 ⊢ ((𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐵) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺) → 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺))) |
25 | 12, 13, 24 | syl2anc 584 | . 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 1541 ∈ wcel 2106 ∀wral 3062 ∩ cin 3907 class class class wbr 5103 ran crn 5632 ‘cfv 6493 〈“cs3 14723 Basecbs 17075 distcds 17134 TarskiGcstrkg 27255 Itvcitv 27261 LineGclng 27262 ∟Gcrag 27521 ⟂Gcperpg 27523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-oadd 8412 df-er 8644 df-map 8763 df-pm 8764 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-dju 9833 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-n0 12410 df-xnn0 12482 df-z 12496 df-uz 12760 df-fz 13417 df-fzo 13560 df-hash 14223 df-word 14395 df-concat 14451 df-s1 14476 df-s2 14729 df-s3 14730 df-trkgc 27276 df-trkgb 27277 df-trkgcb 27278 df-trkg 27281 df-cgrg 27339 df-mir 27481 df-rag 27522 df-perpg 27524 |
This theorem is referenced by: perprag 27554 |
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