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Theorem mireq 25781
Description: Equality deduction for point inversion. Theorem 7.9 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-May-2019.)
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)
mirval.a (𝜑𝐴𝑃)
mirfv.m 𝑀 = (𝑆𝐴)
mirmir.b (𝜑𝐵𝑃)
mireq.c (𝜑𝐶𝑃)
mireq.d (𝜑 → (𝑀𝐵) = (𝑀𝐶))
Assertion
Ref Expression
mireq (𝜑𝐵 = 𝐶)

Proof of Theorem mireq
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mirval.p . . 3 𝑃 = (Base‘𝐺)
2 mirval.d . . 3 = (dist‘𝐺)
3 mirval.i . . 3 𝐼 = (Itv‘𝐺)
4 mirval.l . . 3 𝐿 = (LineG‘𝐺)
5 mirval.s . . 3 𝑆 = (pInvG‘𝐺)
6 mirval.g . . 3 (𝜑𝐺 ∈ TarskiG)
7 mirval.a . . 3 (𝜑𝐴𝑃)
8 mirfv.m . . 3 𝑀 = (𝑆𝐴)
9 mireq.c . . . 4 (𝜑𝐶𝑃)
101, 2, 3, 4, 5, 6, 7, 8, 9mircl 25777 . . 3 (𝜑 → (𝑀𝐶) ∈ 𝑃)
11 mirmir.b . . 3 (𝜑𝐵𝑃)
121, 2, 3, 4, 5, 6, 7, 8, 11mirfv 25772 . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
13 mireq.d . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑀𝐶))
1412, 13eqtr3d 2807 . . . . . 6 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀𝐶))
151, 2, 3, 6, 11, 7mirreu3 25770 . . . . . . 7 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
16 oveq2 6804 . . . . . . . . . 10 (𝑧 = (𝑀𝐶) → (𝐴 𝑧) = (𝐴 (𝑀𝐶)))
1716eqeq1d 2773 . . . . . . . . 9 (𝑧 = (𝑀𝐶) → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 (𝑀𝐶)) = (𝐴 𝐵)))
18 oveq1 6803 . . . . . . . . . 10 (𝑧 = (𝑀𝐶) → (𝑧𝐼𝐵) = ((𝑀𝐶)𝐼𝐵))
1918eleq2d 2836 . . . . . . . . 9 (𝑧 = (𝑀𝐶) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵)))
2017, 19anbi12d 616 . . . . . . . 8 (𝑧 = (𝑀𝐶) → (((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 (𝑀𝐶)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵))))
2120riota2 6779 . . . . . . 7 (((𝑀𝐶) ∈ 𝑃 ∧ ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) → (((𝐴 (𝑀𝐶)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵)) ↔ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀𝐶)))
2210, 15, 21syl2anc 573 . . . . . 6 (𝜑 → (((𝐴 (𝑀𝐶)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵)) ↔ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀𝐶)))
2314, 22mpbird 247 . . . . 5 (𝜑 → ((𝐴 (𝑀𝐶)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵)))
2423simpld 482 . . . 4 (𝜑 → (𝐴 (𝑀𝐶)) = (𝐴 𝐵))
2524eqcomd 2777 . . 3 (𝜑 → (𝐴 𝐵) = (𝐴 (𝑀𝐶)))
2623simprd 483 . . . 4 (𝜑𝐴 ∈ ((𝑀𝐶)𝐼𝐵))
271, 2, 3, 6, 10, 7, 11, 26tgbtwncom 25604 . . 3 (𝜑𝐴 ∈ (𝐵𝐼(𝑀𝐶)))
281, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27ismir 25775 . 2 (𝜑𝐵 = (𝑀‘(𝑀𝐶)))
291, 2, 3, 4, 5, 6, 7, 8, 9mirmir 25778 . 2 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
3028, 29eqtrd 2805 1 (𝜑𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  ∃!wreu 3063  cfv 6030  crio 6756  (class class class)co 6796  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557  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 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-mir 25769
This theorem is referenced by:  mirhl  25795  mirbtwnhl  25796  mirhl2  25797  colperpexlem3  25845
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