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Theorem mireq 25477
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 25473 . . 3 (𝜑 → (𝑀𝐶) ∈ 𝑃)
11 mirmir.b . . 3 (𝜑𝐵𝑃)
121, 2, 3, 4, 5, 6, 7, 8, 11mirfv 25468 . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
13 mireq.d . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑀𝐶))
1412, 13eqtr3d 2657 . . . . . 6 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀𝐶))
151, 2, 3, 6, 11, 7mirreu3 25466 . . . . . . 7 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
16 oveq2 6618 . . . . . . . . . 10 (𝑧 = (𝑀𝐶) → (𝐴 𝑧) = (𝐴 (𝑀𝐶)))
1716eqeq1d 2623 . . . . . . . . 9 (𝑧 = (𝑀𝐶) → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 (𝑀𝐶)) = (𝐴 𝐵)))
18 oveq1 6617 . . . . . . . . . 10 (𝑧 = (𝑀𝐶) → (𝑧𝐼𝐵) = ((𝑀𝐶)𝐼𝐵))
1918eleq2d 2684 . . . . . . . . 9 (𝑧 = (𝑀𝐶) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵)))
2017, 19anbi12d 746 . . . . . . . 8 (𝑧 = (𝑀𝐶) → (((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 (𝑀𝐶)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵))))
2120riota2 6593 . . . . . . 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 692 . . . . . 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 475 . . . 4 (𝜑 → (𝐴 (𝑀𝐶)) = (𝐴 𝐵))
2524eqcomd 2627 . . 3 (𝜑 → (𝐴 𝐵) = (𝐴 (𝑀𝐶)))
2623simprd 479 . . . 4 (𝜑𝐴 ∈ ((𝑀𝐶)𝐼𝐵))
271, 2, 3, 6, 10, 7, 11, 26tgbtwncom 25300 . . 3 (𝜑𝐴 ∈ (𝐵𝐼(𝑀𝐶)))
281, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27ismir 25471 . 2 (𝜑𝐵 = (𝑀‘(𝑀𝐶)))
291, 2, 3, 4, 5, 6, 7, 8, 9mirmir 25474 . 2 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
3028, 29eqtrd 2655 1 (𝜑𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  ∃!wreu 2909  cfv 5852  crio 6570  (class class class)co 6610  Basecbs 15792  distcds 15882  TarskiGcstrkg 25246  Itvcitv 25252  LineGclng 25253  pInvGcmir 25464
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  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 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-trkgc 25264  df-trkgb 25265  df-trkgcb 25266  df-trkg 25269  df-mir 25465
This theorem is referenced by:  mirhl  25491  mirbtwnhl  25492  mirhl2  25493  colperpexlem3  25541
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