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Mirrors > Home > MPE Home > Th. List > mideulem | Structured version Visualization version GIF version |
Description: Lemma for mideu 27095. We can assume mideulem.9 "without loss of generality". (Contributed by Thierry Arnoux, 25-Nov-2019.) |
Ref | Expression |
---|---|
colperpex.p | ⊢ 𝑃 = (Base‘𝐺) |
colperpex.d | ⊢ − = (dist‘𝐺) |
colperpex.i | ⊢ 𝐼 = (Itv‘𝐺) |
colperpex.l | ⊢ 𝐿 = (LineG‘𝐺) |
colperpex.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
mideu.s | ⊢ 𝑆 = (pInvG‘𝐺) |
mideu.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
mideu.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
mideulem.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
mideulem.2 | ⊢ (𝜑 → 𝑄 ∈ 𝑃) |
mideulem.3 | ⊢ (𝜑 → 𝑂 ∈ 𝑃) |
mideulem.4 | ⊢ (𝜑 → 𝑇 ∈ 𝑃) |
mideulem.5 | ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵)) |
mideulem.6 | ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂)) |
mideulem.7 | ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵)) |
mideulem.8 | ⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂)) |
mideulem.9 | ⊢ (𝜑 → (𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄)) |
Ref | Expression |
---|---|
mideulem | ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprrl 778 | . . 3 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) ∧ (𝑥 ∈ 𝑃 ∧ (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑟)))) → 𝐵 = ((𝑆‘𝑥)‘𝐴)) | |
2 | colperpex.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
3 | colperpex.d | . . . 4 ⊢ − = (dist‘𝐺) | |
4 | colperpex.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | colperpex.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | colperpex.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | 6 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐺 ∈ TarskiG) |
8 | mideu.s | . . . 4 ⊢ 𝑆 = (pInvG‘𝐺) | |
9 | mideu.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | 9 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐴 ∈ 𝑃) |
11 | mideu.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
12 | 11 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐵 ∈ 𝑃) |
13 | mideulem.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
14 | 13 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝐴 ≠ 𝐵) |
15 | mideulem.2 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝑃) | |
16 | 15 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑄 ∈ 𝑃) |
17 | mideulem.3 | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑃) | |
18 | 17 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑂 ∈ 𝑃) |
19 | mideulem.4 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑃) | |
20 | 19 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ 𝑃) |
21 | mideulem.5 | . . . . 5 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵)) | |
22 | 21 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵)) |
23 | mideulem.6 | . . . . 5 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂)) | |
24 | 23 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂)) |
25 | mideulem.7 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝐴𝐿𝐵)) | |
26 | 25 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ (𝐴𝐿𝐵)) |
27 | mideulem.8 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (𝑄𝐼𝑂)) | |
28 | 27 | ad2antrr 723 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑇 ∈ (𝑄𝐼𝑂)) |
29 | simplr 766 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑟 ∈ 𝑃) | |
30 | simprl 768 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → 𝑟 ∈ (𝐵𝐼𝑄)) | |
31 | simprr 770 | . . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → (𝐴 − 𝑂) = (𝐵 − 𝑟)) | |
32 | 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 29, 30, 31 | opphllem 27092 | . . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → ∃𝑥 ∈ 𝑃 (𝐵 = ((𝑆‘𝑥)‘𝐴) ∧ 𝑂 = ((𝑆‘𝑥)‘𝑟))) |
33 | 1, 32 | reximddv 3206 | . 2 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴)) |
34 | mideulem.9 | . . 3 ⊢ (𝜑 → (𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄)) | |
35 | eqid 2740 | . . . 4 ⊢ (≤G‘𝐺) = (≤G‘𝐺) | |
36 | 2, 3, 4, 35, 6, 9, 17, 11, 15 | legov 26942 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝑂)(≤G‘𝐺)(𝐵 − 𝑄) ↔ ∃𝑟 ∈ 𝑃 (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟)))) |
37 | 34, 36 | mpbid 231 | . 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑃 (𝑟 ∈ (𝐵𝐼𝑄) ∧ (𝐴 − 𝑂) = (𝐵 − 𝑟))) |
38 | 33, 37 | r19.29a 3220 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝐵 = ((𝑆‘𝑥)‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∃wrex 3067 class class class wbr 5079 ‘cfv 6431 (class class class)co 7269 Basecbs 16908 distcds 16967 TarskiGcstrkg 26784 Itvcitv 26790 LineGclng 26791 ≤Gcleg 26939 pInvGcmir 27009 ⟂Gcperpg 27052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-oadd 8290 df-er 8479 df-map 8598 df-pm 8599 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-dju 9658 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-n0 12232 df-xnn0 12304 df-z 12318 df-uz 12580 df-fz 13237 df-fzo 13380 df-hash 14041 df-word 14214 df-concat 14270 df-s1 14297 df-s2 14557 df-s3 14558 df-trkgc 26805 df-trkgb 26806 df-trkgcb 26807 df-trkg 26810 df-cgrg 26868 df-leg 26940 df-mir 27010 df-rag 27051 df-perpg 27053 |
This theorem is referenced by: midex 27094 |
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