Theorem mireq 28751

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 28747	. . 3 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃)
11		mirmir.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	1, 2, 3, 4, 5, 6, 7, 8, 11	mirfv 28742	. . . . . . 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 2776	. . . . . 6 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀‘𝐶))
15	1, 2, 3, 6, 11, 7	mirreu3 28740	. . . . . . 7 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
16		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑧 = (𝑀‘𝐶) → (𝐴 − 𝑧) = (𝐴 − (𝑀‘𝐶)))
17	16	eqeq1d 2741	. . . . . . . . 9 ⊢ (𝑧 = (𝑀‘𝐶) → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵)))
18		oveq1 7363	. . . . . . . . . 10 ⊢ (𝑧 = (𝑀‘𝐶) → (𝑧𝐼𝐵) = ((𝑀‘𝐶)𝐼𝐵))
19	18	eleq2d 2825	. . . . . . . . 9 ⊢ (𝑧 = (𝑀‘𝐶) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵)))
20	17, 19	anbi12d 638	. . . . . . . 8 ⊢ (𝑧 = (𝑀‘𝐶) → (((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵))))
21	20	riota2 7338	. . . . . . 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 590	. . . . . 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 258	. . . . 5 ⊢ (𝜑 → ((𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵)))
24	23	simpld 495	. . . 4 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵))
25	24	eqcomd 2745	. . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − (𝑀‘𝐶)))
26	23	simprd 496	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵))
27	1, 2, 3, 6, 10, 7, 11, 26	tgbtwncom 28574	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼(𝑀‘𝐶)))
28	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27	ismir 28745	. 2 ⊢ (𝜑 → 𝐵 = (𝑀‘(𝑀‘𝐶)))
29	1, 2, 3, 4, 5, 6, 7, 8, 9	mirmir 28748	. 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶)
30	28, 29	eqtrd 2774	1 ⊢ (𝜑 → 𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃!wreu 3342  ‘cfv 6485  ℩crio 7312  (class class class)co 7356  Basecbs 17170  distcds 17220  TarskiGcstrkg 28513  Itvcitv 28519  LineGclng 28520  pInvGcmir 28738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-trkgc 28534  df-trkgb 28535  df-trkgcb 28536  df-trkg 28539  df-mir 28739

This theorem is referenced by:  mirhl  28765  mirbtwnhl  28766  colperpexlem3  28818