Theorem mireq 28636

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirval.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirfv.m	⊢ 𝑀 = (𝑆‘𝐴)
mirmir.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mireq.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
mireq.d	⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘𝐶))

Assertion

Ref	Expression
mireq	⊢ (𝜑 → 𝐵 = 𝐶)

Proof of Theorem mireq

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		mirval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		mirfv.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐴)
9		mireq.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mircl 28632	. . 3 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃)
11		mirmir.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	1, 2, 3, 4, 5, 6, 7, 8, 11	mirfv 28627	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
13		mireq.d	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐵) = (𝑀‘𝐶))
14	12, 13	eqtr3d 2767	. . . . . 6 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀‘𝐶))
15	1, 2, 3, 6, 11, 7	mirreu3 28625	. . . . . . 7 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
16		oveq2 7349	. . . . . . . . . 10 ⊢ (𝑧 = (𝑀‘𝐶) → (𝐴 − 𝑧) = (𝐴 − (𝑀‘𝐶)))
17	16	eqeq1d 2732	. . . . . . . . 9 ⊢ (𝑧 = (𝑀‘𝐶) → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵)))
18		oveq1 7348	. . . . . . . . . 10 ⊢ (𝑧 = (𝑀‘𝐶) → (𝑧𝐼𝐵) = ((𝑀‘𝐶)𝐼𝐵))
19	18	eleq2d 2815	. . . . . . . . 9 ⊢ (𝑧 = (𝑀‘𝐶) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵)))
20	17, 19	anbi12d 632	. . . . . . . 8 ⊢ (𝑧 = (𝑀‘𝐶) → (((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵))))
21	20	riota2 7323	. . . . . . 7 ⊢ (((𝑀‘𝐶) ∈ 𝑃 ∧ ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) → (((𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵)) ↔ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀‘𝐶)))
22	10, 15, 21	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (((𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵)) ↔ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀‘𝐶)))
23	14, 22	mpbird 257	. . . . 5 ⊢ (𝜑 → ((𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵)))
24	23	simpld 494	. . . 4 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵))
25	24	eqcomd 2736	. . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − (𝑀‘𝐶)))
26	23	simprd 495	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵))
27	1, 2, 3, 6, 10, 7, 11, 26	tgbtwncom 28459	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼(𝑀‘𝐶)))
28	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27	ismir 28630	. 2 ⊢ (𝜑 → 𝐵 = (𝑀‘(𝑀‘𝐶)))
29	1, 2, 3, 4, 5, 6, 7, 8, 9	mirmir 28633	. 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶)
30	28, 29	eqtrd 2765	1 ⊢ (𝜑 → 𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∃!wreu 3342  ‘cfv 6477  ℩crio 7297  (class class class)co 7341  Basecbs 17112  distcds 17162  TarskiGcstrkg 28398  Itvcitv 28404  LineGclng 28405  pInvGcmir 28623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-trkgc 28419  df-trkgb 28420  df-trkgcb 28421  df-trkg 28424  df-mir 28624

This theorem is referenced by:  mirhl  28650  mirbtwnhl  28651  colperpexlem3  28703