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Theorem mircgr 28412
Description: Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Baseβ€˜πΊ)
mirval.d βˆ’ = (distβ€˜πΊ)
mirval.i 𝐼 = (Itvβ€˜πΊ)
mirval.l 𝐿 = (LineGβ€˜πΊ)
mirval.s 𝑆 = (pInvGβ€˜πΊ)
mirval.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
mirval.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
mirfv.m 𝑀 = (π‘†β€˜π΄)
mirfv.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
Assertion
Ref Expression
mircgr (πœ‘ β†’ (𝐴 βˆ’ (π‘€β€˜π΅)) = (𝐴 βˆ’ 𝐡))

Proof of Theorem mircgr
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mirval.p . . . . 5 𝑃 = (Baseβ€˜πΊ)
2 mirval.d . . . . 5 βˆ’ = (distβ€˜πΊ)
3 mirval.i . . . . 5 𝐼 = (Itvβ€˜πΊ)
4 mirval.l . . . . 5 𝐿 = (LineGβ€˜πΊ)
5 mirval.s . . . . 5 𝑆 = (pInvGβ€˜πΊ)
6 mirval.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ TarskiG)
7 mirval.a . . . . 5 (πœ‘ β†’ 𝐴 ∈ 𝑃)
8 mirfv.m . . . . 5 𝑀 = (π‘†β€˜π΄)
9 mirfv.b . . . . 5 (πœ‘ β†’ 𝐡 ∈ 𝑃)
101, 2, 3, 4, 5, 6, 7, 8, 9mirfv 28411 . . . 4 (πœ‘ β†’ (π‘€β€˜π΅) = (℩𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ (𝑧𝐼𝐡))))
111, 2, 3, 6, 9, 7mirreu3 28409 . . . . 5 (πœ‘ β†’ βˆƒ!𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ (𝑧𝐼𝐡)))
12 riotacl2 7377 . . . . 5 (βˆƒ!𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ (𝑧𝐼𝐡)) β†’ (℩𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ (𝑧𝐼𝐡))) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ (𝑧𝐼𝐡))})
1311, 12syl 17 . . . 4 (πœ‘ β†’ (℩𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ (𝑧𝐼𝐡))) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ (𝑧𝐼𝐡))})
1410, 13eqeltrd 2827 . . 3 (πœ‘ β†’ (π‘€β€˜π΅) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ (𝑧𝐼𝐡))})
15 oveq2 7412 . . . . . 6 (𝑧 = (π‘€β€˜π΅) β†’ (𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ (π‘€β€˜π΅)))
1615eqeq1d 2728 . . . . 5 (𝑧 = (π‘€β€˜π΅) β†’ ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ↔ (𝐴 βˆ’ (π‘€β€˜π΅)) = (𝐴 βˆ’ 𝐡)))
17 oveq1 7411 . . . . . 6 (𝑧 = (π‘€β€˜π΅) β†’ (𝑧𝐼𝐡) = ((π‘€β€˜π΅)𝐼𝐡))
1817eleq2d 2813 . . . . 5 (𝑧 = (π‘€β€˜π΅) β†’ (𝐴 ∈ (𝑧𝐼𝐡) ↔ 𝐴 ∈ ((π‘€β€˜π΅)𝐼𝐡)))
1916, 18anbi12d 630 . . . 4 (𝑧 = (π‘€β€˜π΅) β†’ (((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ (𝑧𝐼𝐡)) ↔ ((𝐴 βˆ’ (π‘€β€˜π΅)) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ ((π‘€β€˜π΅)𝐼𝐡))))
2019elrab 3678 . . 3 ((π‘€β€˜π΅) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ (𝑧𝐼𝐡))} ↔ ((π‘€β€˜π΅) ∈ 𝑃 ∧ ((𝐴 βˆ’ (π‘€β€˜π΅)) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ ((π‘€β€˜π΅)𝐼𝐡))))
2114, 20sylib 217 . 2 (πœ‘ β†’ ((π‘€β€˜π΅) ∈ 𝑃 ∧ ((𝐴 βˆ’ (π‘€β€˜π΅)) = (𝐴 βˆ’ 𝐡) ∧ 𝐴 ∈ ((π‘€β€˜π΅)𝐼𝐡))))
2221simprld 769 1 (πœ‘ β†’ (𝐴 βˆ’ (π‘€β€˜π΅)) = (𝐴 βˆ’ 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆƒ!wreu 3368  {crab 3426  β€˜cfv 6536  β„©crio 7359  (class class class)co 7404  Basecbs 17151  distcds 17213  TarskiGcstrkg 28182  Itvcitv 28188  LineGclng 28189  pInvGcmir 28407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-trkgc 28203  df-trkgb 28204  df-trkgcb 28205  df-trkg 28208  df-mir 28408
This theorem is referenced by:  mirmir  28417  miriso  28425  mirmir2  28429  mircgrextend  28437  mirtrcgr  28438  mirauto  28439  miduniq  28440  krippenlem  28445  ragcol  28454  ragflat  28459  ragcgr  28462  footexALT  28473  footexlem2  28475  colperpexlem1  28485  colperpexlem3  28487  mideulem2  28489  opphllem  28490  midcgr  28535  lmiisolem  28551
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