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Theorem mircgrextend 25472
Description: Link congruence over a pair of mirror points. cf tgcgrextend 25275. (Contributed by Thierry Arnoux, 4-Oct-2020.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Base‘𝐺)
mirval.d = (dist‘𝐺)
mirval.i 𝐼 = (Itv‘𝐺)
mirval.l 𝐿 = (LineG‘𝐺)
mirval.s 𝑆 = (pInvG‘𝐺)
mirval.g (𝜑𝐺 ∈ TarskiG)
mirtrcgr.e = (cgrG‘𝐺)
mirtrcgr.m 𝑀 = (𝑆𝐵)
mirtrcgr.n 𝑁 = (𝑆𝑌)
mirtrcgr.a (𝜑𝐴𝑃)
mirtrcgr.b (𝜑𝐵𝑃)
mirtrcgr.x (𝜑𝑋𝑃)
mirtrcgr.y (𝜑𝑌𝑃)
mircgrextend.1 (𝜑 → (𝐴 𝐵) = (𝑋 𝑌))
Assertion
Ref Expression
mircgrextend (𝜑 → (𝐴 (𝑀𝐴)) = (𝑋 (𝑁𝑋)))

Proof of Theorem mircgrextend
StepHypRef Expression
1 mirval.p . 2 𝑃 = (Base‘𝐺)
2 mirval.d . 2 = (dist‘𝐺)
3 mirval.i . 2 𝐼 = (Itv‘𝐺)
4 mirval.g . 2 (𝜑𝐺 ∈ TarskiG)
5 mirtrcgr.a . 2 (𝜑𝐴𝑃)
6 mirtrcgr.b . 2 (𝜑𝐵𝑃)
7 mirval.l . . 3 𝐿 = (LineG‘𝐺)
8 mirval.s . . 3 𝑆 = (pInvG‘𝐺)
9 mirtrcgr.m . . 3 𝑀 = (𝑆𝐵)
101, 2, 3, 7, 8, 4, 6, 9, 5mircl 25451 . 2 (𝜑 → (𝑀𝐴) ∈ 𝑃)
11 mirtrcgr.x . 2 (𝜑𝑋𝑃)
12 mirtrcgr.y . 2 (𝜑𝑌𝑃)
13 mirtrcgr.n . . 3 𝑁 = (𝑆𝑌)
141, 2, 3, 7, 8, 4, 12, 13, 11mircl 25451 . 2 (𝜑 → (𝑁𝑋) ∈ 𝑃)
151, 2, 3, 7, 8, 4, 6, 9, 5mirbtwn 25448 . . 3 (𝜑𝐵 ∈ ((𝑀𝐴)𝐼𝐴))
161, 2, 3, 4, 10, 6, 5, 15tgbtwncom 25278 . 2 (𝜑𝐵 ∈ (𝐴𝐼(𝑀𝐴)))
171, 2, 3, 7, 8, 4, 12, 13, 11mirbtwn 25448 . . 3 (𝜑𝑌 ∈ ((𝑁𝑋)𝐼𝑋))
181, 2, 3, 4, 14, 12, 11, 17tgbtwncom 25278 . 2 (𝜑𝑌 ∈ (𝑋𝐼(𝑁𝑋)))
19 mircgrextend.1 . 2 (𝜑 → (𝐴 𝐵) = (𝑋 𝑌))
201, 2, 3, 4, 5, 6, 11, 12, 19tgcgrcomlr 25270 . . 3 (𝜑 → (𝐵 𝐴) = (𝑌 𝑋))
211, 2, 3, 7, 8, 4, 6, 9, 5mircgr 25447 . . 3 (𝜑 → (𝐵 (𝑀𝐴)) = (𝐵 𝐴))
221, 2, 3, 7, 8, 4, 12, 13, 11mircgr 25447 . . 3 (𝜑 → (𝑌 (𝑁𝑋)) = (𝑌 𝑋))
2320, 21, 223eqtr4d 2670 . 2 (𝜑 → (𝐵 (𝑀𝐴)) = (𝑌 (𝑁𝑋)))
241, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23tgcgrextend 25275 1 (𝜑 → (𝐴 (𝑀𝐴)) = (𝑋 (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  cfv 5850  (class class class)co 6605  Basecbs 15776  distcds 15866  TarskiGcstrkg 25224  Itvcitv 25230  LineGclng 25231  cgrGccgrg 25300  pInvGcmir 25442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-trkgc 25242  df-trkgb 25243  df-trkgcb 25244  df-trkg 25247  df-mir 25443
This theorem is referenced by:  mirtrcgr  25473
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