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Mirrors > Home > MPE Home > Th. List > mircgrextend | Structured version Visualization version GIF version |
Description: Link congruence over a pair of mirror points. cf tgcgrextend 25180. (Contributed by Thierry Arnoux, 4-Oct-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) |
mirtrcgr.e | ⊢ ∼ = (cgrG‘𝐺) |
mirtrcgr.m | ⊢ 𝑀 = (𝑆‘𝐵) |
mirtrcgr.n | ⊢ 𝑁 = (𝑆‘𝑌) |
mirtrcgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
mirtrcgr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
mirtrcgr.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
mirtrcgr.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
mircgrextend.1 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌)) |
Ref | Expression |
---|---|
mircgrextend | ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | mirtrcgr.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | mirtrcgr.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
9 | mirtrcgr.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐵) | |
10 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircl 25356 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
11 | mirtrcgr.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
12 | mirtrcgr.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
13 | mirtrcgr.n | . . 3 ⊢ 𝑁 = (𝑆‘𝑌) | |
14 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircl 25356 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑃) |
15 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mirbtwn 25353 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝑀‘𝐴)𝐼𝐴)) |
16 | 1, 2, 3, 4, 10, 6, 5, 15 | tgbtwncom 25183 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼(𝑀‘𝐴))) |
17 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mirbtwn 25353 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ((𝑁‘𝑋)𝐼𝑋)) |
18 | 1, 2, 3, 4, 14, 12, 11, 17 | tgbtwncom 25183 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼(𝑁‘𝑋))) |
19 | mircgrextend.1 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌)) | |
20 | 1, 2, 3, 4, 5, 6, 11, 12, 19 | tgcgrcomlr 25175 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝑌 − 𝑋)) |
21 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircgr 25352 | . . 3 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝐵 − 𝐴)) |
22 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircgr 25352 | . . 3 ⊢ (𝜑 → (𝑌 − (𝑁‘𝑋)) = (𝑌 − 𝑋)) |
23 | 20, 21, 22 | 3eqtr4d 2654 | . 2 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝑌 − (𝑁‘𝑋))) |
24 | 1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23 | tgcgrextend 25180 | 1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 LineGclng 25136 cgrGccgrg 25205 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: mirtrcgr 25378 |
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