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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 26455 | . . 3 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃) |
11 | mirmir.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 11 | mirfv 26450 | . . . . . . 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 2835 | . . . . . 6 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = (𝑀‘𝐶)) |
15 | 1, 2, 3, 6, 11, 7 | mirreu3 26448 | . . . . . . 7 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) |
16 | oveq2 7143 | . . . . . . . . . 10 ⊢ (𝑧 = (𝑀‘𝐶) → (𝐴 − 𝑧) = (𝐴 − (𝑀‘𝐶))) | |
17 | 16 | eqeq1d 2800 | . . . . . . . . 9 ⊢ (𝑧 = (𝑀‘𝐶) → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵))) |
18 | oveq1 7142 | . . . . . . . . . 10 ⊢ (𝑧 = (𝑀‘𝐶) → (𝑧𝐼𝐵) = ((𝑀‘𝐶)𝐼𝐵)) | |
19 | 18 | eleq2d 2875 | . . . . . . . . 9 ⊢ (𝑧 = (𝑀‘𝐶) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵))) |
20 | 17, 19 | anbi12d 633 | . . . . . . . 8 ⊢ (𝑧 = (𝑀‘𝐶) → (((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵)))) |
21 | 20 | riota2 7118 | . . . . . . 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 587 | . . . . . 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 260 | . . . . 5 ⊢ (𝜑 → ((𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵))) |
24 | 23 | simpld 498 | . . . 4 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐶)) = (𝐴 − 𝐵)) |
25 | 24 | eqcomd 2804 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − (𝑀‘𝐶))) |
26 | 23 | simprd 499 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐶)𝐼𝐵)) |
27 | 1, 2, 3, 6, 10, 7, 11, 26 | tgbtwncom 26282 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼(𝑀‘𝐶))) |
28 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27 | ismir 26453 | . 2 ⊢ (𝜑 → 𝐵 = (𝑀‘(𝑀‘𝐶))) |
29 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mirmir 26456 | . 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶) |
30 | 28, 29 | eqtrd 2833 | 1 ⊢ (𝜑 → 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃!wreu 3108 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231 pInvGcmir 26446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 df-mir 26447 |
This theorem is referenced by: mirhl 26473 mirbtwnhl 26474 colperpexlem3 26526 |
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