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Theorem mireq 26443
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 26439 . . 3 (𝜑 → (𝑀𝐶) ∈ 𝑃)
11 mirmir.b . . 3 (𝜑𝐵𝑃)
121, 2, 3, 4, 5, 6, 7, 8, 11mirfv 26434 . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
13 mireq.d . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑀𝐶))
1412, 13eqtr3d 2856 . . . . . 6 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀𝐶))
151, 2, 3, 6, 11, 7mirreu3 26432 . . . . . . 7 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
16 oveq2 7156 . . . . . . . . . 10 (𝑧 = (𝑀𝐶) → (𝐴 𝑧) = (𝐴 (𝑀𝐶)))
1716eqeq1d 2821 . . . . . . . . 9 (𝑧 = (𝑀𝐶) → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 (𝑀𝐶)) = (𝐴 𝐵)))
18 oveq1 7155 . . . . . . . . . 10 (𝑧 = (𝑀𝐶) → (𝑧𝐼𝐵) = ((𝑀𝐶)𝐼𝐵))
1918eleq2d 2896 . . . . . . . . 9 (𝑧 = (𝑀𝐶) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵)))
2017, 19anbi12d 632 . . . . . . . 8 (𝑧 = (𝑀𝐶) → (((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 (𝑀𝐶)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵))))
2120riota2 7131 . . . . . . 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 586 . . . . . 6 (𝜑 → (((𝐴 (𝑀𝐶)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵)) ↔ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀𝐶)))
2314, 22mpbird 259 . . . . 5 (𝜑 → ((𝐴 (𝑀𝐶)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵)))
2423simpld 497 . . . 4 (𝜑 → (𝐴 (𝑀𝐶)) = (𝐴 𝐵))
2524eqcomd 2825 . . 3 (𝜑 → (𝐴 𝐵) = (𝐴 (𝑀𝐶)))
2623simprd 498 . . . 4 (𝜑𝐴 ∈ ((𝑀𝐶)𝐼𝐵))
271, 2, 3, 6, 10, 7, 11, 26tgbtwncom 26266 . . 3 (𝜑𝐴 ∈ (𝐵𝐼(𝑀𝐶)))
281, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27ismir 26437 . 2 (𝜑𝐵 = (𝑀‘(𝑀𝐶)))
291, 2, 3, 4, 5, 6, 7, 8, 9mirmir 26440 . 2 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
3028, 29eqtrd 2854 1 (𝜑𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  ∃!wreu 3138  cfv 6348  crio 7105  (class class class)co 7148  Basecbs 16475  distcds 16566  TarskiGcstrkg 26208  Itvcitv 26214  LineGclng 26215  pInvGcmir 26430
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-trkgc 26226  df-trkgb 26227  df-trkgcb 26228  df-trkg 26231  df-mir 26431
This theorem is referenced by:  mirhl  26457  mirbtwnhl  26458  colperpexlem3  26510
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