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Mirrors > Home > MPE Home > Th. List > mirne | Structured version Visualization version GIF version |
Description: Mirror of non-center point cannot be the center point. (Contributed by Thierry Arnoux, 27-Sep-2020.) |
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) |
mirinv.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
mirne.1 | ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
Ref | Expression |
---|---|
mirne | ⊢ (𝜑 → (𝑀‘𝐵) ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘𝐵) = 𝐴) | |
2 | 1 | fveq2d 6892 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘(𝑀‘𝐵)) = (𝑀‘𝐴)) |
3 | mirval.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
4 | mirval.d | . . . . . 6 ⊢ − = (dist‘𝐺) | |
5 | mirval.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
6 | mirval.l | . . . . . 6 ⊢ 𝐿 = (LineG‘𝐺) | |
7 | mirval.s | . . . . . 6 ⊢ 𝑆 = (pInvG‘𝐺) | |
8 | mirval.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
9 | mirval.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | mirfv.m | . . . . . 6 ⊢ 𝑀 = (𝑆‘𝐴) | |
11 | mirinv.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
12 | 3, 4, 5, 6, 7, 8, 9, 10, 11 | mirmir 27902 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵) |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘(𝑀‘𝐵)) = 𝐵) |
14 | eqid 2732 | . . . . . 6 ⊢ 𝐴 = 𝐴 | |
15 | 3, 4, 5, 6, 7, 8, 9, 10, 9 | mirinv 27906 | . . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 = 𝐴)) |
16 | 14, 15 | mpbiri 257 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘𝐴) = 𝐴) |
18 | 2, 13, 17 | 3eqtr3d 2780 | . . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → 𝐵 = 𝐴) |
19 | mirne.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴) | |
20 | 19 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → 𝐵 ≠ 𝐴) |
21 | 20 | neneqd 2945 | . . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → ¬ 𝐵 = 𝐴) |
22 | 18, 21 | pm2.65da 815 | . 2 ⊢ (𝜑 → ¬ (𝑀‘𝐵) = 𝐴) |
23 | 22 | neqned 2947 | 1 ⊢ (𝜑 → (𝑀‘𝐵) ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ‘cfv 6540 Basecbs 17140 distcds 17202 TarskiGcstrkg 27667 Itvcitv 27673 LineGclng 27674 pInvGcmir 27892 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-trkgc 27688 df-trkgb 27689 df-trkgcb 27690 df-trkg 27693 df-mir 27893 |
This theorem is referenced by: mirhl2 27921 sacgr 28071 |
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