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Theorem mireq 28688
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 28684 . . 3 (𝜑 → (𝑀𝐶) ∈ 𝑃)
11 mirmir.b . . 3 (𝜑𝐵𝑃)
121, 2, 3, 4, 5, 6, 7, 8, 11mirfv 28679 . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
13 mireq.d . . . . . . 7 (𝜑 → (𝑀𝐵) = (𝑀𝐶))
1412, 13eqtr3d 2777 . . . . . 6 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀𝐶))
151, 2, 3, 6, 11, 7mirreu3 28677 . . . . . . 7 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
16 oveq2 7439 . . . . . . . . . 10 (𝑧 = (𝑀𝐶) → (𝐴 𝑧) = (𝐴 (𝑀𝐶)))
1716eqeq1d 2737 . . . . . . . . 9 (𝑧 = (𝑀𝐶) → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 (𝑀𝐶)) = (𝐴 𝐵)))
18 oveq1 7438 . . . . . . . . . 10 (𝑧 = (𝑀𝐶) → (𝑧𝐼𝐵) = ((𝑀𝐶)𝐼𝐵))
1918eleq2d 2825 . . . . . . . . 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 7413 . . . . . . 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 2741 . . 3 (𝜑 → (𝐴 𝐵) = (𝐴 (𝑀𝐶)))
2623simprd 495 . . . 4 (𝜑𝐴 ∈ ((𝑀𝐶)𝐼𝐵))
271, 2, 3, 6, 10, 7, 11, 26tgbtwncom 28511 . . 3 (𝜑𝐴 ∈ (𝐵𝐼(𝑀𝐶)))
281, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27ismir 28682 . 2 (𝜑𝐵 = (𝑀‘(𝑀𝐶)))
291, 2, 3, 4, 5, 6, 7, 8, 9mirmir 28685 . 2 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
3028, 29eqtrd 2775 1 (𝜑𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  ∃!wreu 3376  cfv 6563  crio 7387  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457  pInvGcmir 28675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476  df-mir 28676
This theorem is referenced by:  mirhl  28702  mirbtwnhl  28703  colperpexlem3  28755
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