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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 476 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘𝐵) = 𝐴) | |
2 | 1 | fveq2d 6107 | . . . 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 25357 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵) |
13 | 12 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘(𝑀‘𝐵)) = 𝐵) |
14 | eqid 2610 | . . . . . 6 ⊢ 𝐴 = 𝐴 | |
15 | 3, 4, 5, 6, 7, 8, 9, 10, 9 | mirinv 25361 | . . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 = 𝐴)) |
16 | 14, 15 | mpbiri 247 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴) |
17 | 16 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘𝐴) = 𝐴) |
18 | 2, 13, 17 | 3eqtr3d 2652 | . . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → 𝐵 = 𝐴) |
19 | mirne.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴) | |
20 | 19 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → 𝐵 ≠ 𝐴) |
21 | 20 | neneqd 2787 | . . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → ¬ 𝐵 = 𝐴) |
22 | 18, 21 | pm2.65da 598 | . 2 ⊢ (𝜑 → ¬ (𝑀‘𝐵) = 𝐴) |
23 | 22 | neqned 2789 | 1 ⊢ (𝜑 → (𝑀‘𝐵) ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 LineGclng 25136 pInvGcmir 25347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 df-mir 25348 |
This theorem is referenced by: sacgr 25522 |
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