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Theorem mircgrextend 25798
Description: Link congruence over a pair of mirror points. cf tgcgrextend 25601. (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 25777 . 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 25777 . 2 (𝜑 → (𝑁𝑋) ∈ 𝑃)
151, 2, 3, 7, 8, 4, 6, 9, 5mirbtwn 25774 . . 3 (𝜑𝐵 ∈ ((𝑀𝐴)𝐼𝐴))
161, 2, 3, 4, 10, 6, 5, 15tgbtwncom 25604 . 2 (𝜑𝐵 ∈ (𝐴𝐼(𝑀𝐴)))
171, 2, 3, 7, 8, 4, 12, 13, 11mirbtwn 25774 . . 3 (𝜑𝑌 ∈ ((𝑁𝑋)𝐼𝑋))
181, 2, 3, 4, 14, 12, 11, 17tgbtwncom 25604 . 2 (𝜑𝑌 ∈ (𝑋𝐼(𝑁𝑋)))
19 mircgrextend.1 . 2 (𝜑 → (𝐴 𝐵) = (𝑋 𝑌))
201, 2, 3, 4, 5, 6, 11, 12, 19tgcgrcomlr 25596 . . 3 (𝜑 → (𝐵 𝐴) = (𝑌 𝑋))
211, 2, 3, 7, 8, 4, 6, 9, 5mircgr 25773 . . 3 (𝜑 → (𝐵 (𝑀𝐴)) = (𝐵 𝐴))
221, 2, 3, 7, 8, 4, 12, 13, 11mircgr 25773 . . 3 (𝜑 → (𝑌 (𝑁𝑋)) = (𝑌 𝑋))
2320, 21, 223eqtr4d 2815 . 2 (𝜑 → (𝐵 (𝑀𝐴)) = (𝑌 (𝑁𝑋)))
241, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23tgcgrextend 25601 1 (𝜑 → (𝐴 (𝑀𝐴)) = (𝑋 (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557  cgrGccgrg 25626  pInvGcmir 25768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-mir 25769
This theorem is referenced by:  mirtrcgr  25799
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