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Theorem mireq 28674
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 28670 . . 3 (𝜑 → (𝑀𝐶) ∈ 𝑃)
11 mirmir.b . . 3 (𝜑𝐵𝑃)
121, 2, 3, 4, 5, 6, 7, 8, 11mirfv 28665 . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
13 mireq.d . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑀𝐶))
1412, 13eqtr3d 2778 . . . . . 6 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀𝐶))
151, 2, 3, 6, 11, 7mirreu3 28663 . . . . . . 7 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
16 oveq2 7440 . . . . . . . . . 10 (𝑧 = (𝑀𝐶) → (𝐴 𝑧) = (𝐴 (𝑀𝐶)))
1716eqeq1d 2738 . . . . . . . . 9 (𝑧 = (𝑀𝐶) → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 (𝑀𝐶)) = (𝐴 𝐵)))
18 oveq1 7439 . . . . . . . . . 10 (𝑧 = (𝑀𝐶) → (𝑧𝐼𝐵) = ((𝑀𝐶)𝐼𝐵))
1918eleq2d 2826 . . . . . . . . 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 7414 . . . . . . 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 584 . . . . . 6 (𝜑 → (((𝐴 (𝑀𝐶)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵)) ↔ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀𝐶)))
2314, 22mpbird 257 . . . . 5 (𝜑 → ((𝐴 (𝑀𝐶)) = (𝐴 𝐵) ∧ 𝐴 ∈ ((𝑀𝐶)𝐼𝐵)))
2423simpld 494 . . . 4 (𝜑 → (𝐴 (𝑀𝐶)) = (𝐴 𝐵))
2524eqcomd 2742 . . 3 (𝜑 → (𝐴 𝐵) = (𝐴 (𝑀𝐶)))
2623simprd 495 . . . 4 (𝜑𝐴 ∈ ((𝑀𝐶)𝐼𝐵))
271, 2, 3, 6, 10, 7, 11, 26tgbtwncom 28497 . . 3 (𝜑𝐴 ∈ (𝐵𝐼(𝑀𝐶)))
281, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27ismir 28668 . 2 (𝜑𝐵 = (𝑀‘(𝑀𝐶)))
291, 2, 3, 4, 5, 6, 7, 8, 9mirmir 28671 . 2 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
3028, 29eqtrd 2776 1 (𝜑𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  ∃!wreu 3377  cfv 6560  crio 7388  (class class class)co 7432  Basecbs 17248  distcds 17307  TarskiGcstrkg 28436  Itvcitv 28442  LineGclng 28443  pInvGcmir 28661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-trkgc 28457  df-trkgb 28458  df-trkgcb 28459  df-trkg 28462  df-mir 28662
This theorem is referenced by:  mirhl  28688  mirbtwnhl  28689  colperpexlem3  28741
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