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Mirrors > Home > MPE Home > Th. List > mireq | Structured version Visualization version GIF version |
Description: Equality deduction for point inversion. Theorem 7.9 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-May-2019.) |
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)) |
Ref | Expression |
---|---|
mireq | ⊢ (𝜑 → 𝐵 = 𝐶) |
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 28684 | . . 3 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃) |
11 | mirmir.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 11 | mirfv 28679 | . . . . . . 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 2777 | . . . . . 6 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀‘𝐶)) |
15 | 1, 2, 3, 6, 11, 7 | mirreu3 28677 | . . . . . . 7 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) |
16 | oveq2 7439 | . . . . . . . . . 10 ⊢ (𝑧 = (𝑀‘𝐶) → (𝐴 − 𝑧) = (𝐴 − (𝑀‘𝐶))) | |
17 | 16 | eqeq1d 2737 | . . . . . . . . 9 ⊢ (𝑧 = (𝑀‘𝐶) → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵))) |
18 | oveq1 7438 | . . . . . . . . . 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 632 | . . . . . . . 8 ⊢ (𝑧 = (𝑀‘𝐶) → (((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵)))) |
21 | 20 | riota2 7413 | . . . . . . 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 2741 | . . 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 28511 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼(𝑀‘𝐶))) |
28 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27 | ismir 28682 | . 2 ⊢ (𝜑 → 𝐵 = (𝑀‘(𝑀‘𝐶))) |
29 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mirmir 28685 | . 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶) |
30 | 28, 29 | eqtrd 2775 | 1 ⊢ (𝜑 → 𝐵 = 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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