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Mirrors > Home > MPE Home > Th. List > mirfv | Structured version Visualization version GIF version |
Description: Value of the point inversion function 𝑀. Definition 7.5 of [Schwabhauser] p. 49. (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) |
mirfv.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
mirfv | ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirfv.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐴) | |
2 | mirval.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
3 | mirval.d | . . . 4 ⊢ − = (dist‘𝐺) | |
4 | mirval.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | mirval.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | mirval.s | . . . 4 ⊢ 𝑆 = (pInvG‘𝐺) | |
7 | mirval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
8 | mirval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | 2, 3, 4, 5, 6, 7, 8 | mirval 25550 | . . 3 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))) |
10 | 1, 9 | syl5eq 2668 | . 2 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))) |
11 | simplr 792 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → 𝑦 = 𝐵) | |
12 | 11 | oveq2d 6666 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝐴 − 𝑦) = (𝐴 − 𝐵)) |
13 | 12 | eqeq2d 2632 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → ((𝐴 − 𝑧) = (𝐴 − 𝑦) ↔ (𝐴 − 𝑧) = (𝐴 − 𝐵))) |
14 | 11 | oveq2d 6666 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝑧𝐼𝑦) = (𝑧𝐼𝐵)) |
15 | 14 | eleq2d 2687 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝐴 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝐵))) |
16 | 13, 15 | anbi12d 747 | . . 3 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))) |
17 | 16 | riotabidva 6627 | . 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))) |
18 | mirfv.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
19 | riotaex 6615 | . . 3 ⊢ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ V | |
20 | 19 | a1i 11 | . 2 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ V) |
21 | 10, 17, 18, 20 | fvmptd 6288 | 1 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ↦ cmpt 4729 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 pInvGcmir 25547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-mir 25548 |
This theorem is referenced by: mircgr 25552 mirbtwn 25553 ismir 25554 mirf 25555 mireq 25560 |
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