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Mirrors > Home > MPE Home > Th. List > mirtrcgr | Structured version Visualization version GIF version |
Description: Point inversion of one point of a triangle around another point preserves triangle congruence. (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) |
mirtrcgr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
mirtrcgr.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
mirtrcgr.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
mirtrcgr.2 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝑋𝑌𝑍”〉) |
Ref | Expression |
---|---|
mirtrcgr | ⊢ (𝜑 → 〈“(𝑀‘𝐴)𝐵𝐶”〉 ∼ 〈“(𝑁‘𝑋)𝑌𝑍”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | mirtrcgr.e | . 2 ⊢ ∼ = (cgrG‘𝐺) | |
4 | mirval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | mirval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
6 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
7 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
8 | mirtrcgr.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | mirtrcgr.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐵) | |
10 | mirtrcgr.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 1, 2, 5, 6, 7, 4, 8, 9, 10 | mircl 27879 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
12 | mirtrcgr.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
13 | mirtrcgr.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
14 | mirtrcgr.n | . . 3 ⊢ 𝑁 = (𝑆‘𝑌) | |
15 | mirtrcgr.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
16 | 1, 2, 5, 6, 7, 4, 13, 14, 15 | mircl 27879 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑃) |
17 | mirtrcgr.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
18 | mirtrcgr.2 | . . . . . 6 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝑋𝑌𝑍”〉) | |
19 | 1, 2, 5, 3, 4, 10, 8, 12, 15, 13, 17, 18 | cgr3simp1 27738 | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌)) |
20 | 1, 2, 5, 4, 10, 8, 15, 13, 19 | tgcgrcomlr 27698 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝑌 − 𝑋)) |
21 | 1, 2, 5, 6, 7, 4, 8, 9, 10 | mircgr 27875 | . . . 4 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝐵 − 𝐴)) |
22 | 1, 2, 5, 6, 7, 4, 13, 14, 15 | mircgr 27875 | . . . 4 ⊢ (𝜑 → (𝑌 − (𝑁‘𝑋)) = (𝑌 − 𝑋)) |
23 | 20, 21, 22 | 3eqtr4d 2783 | . . 3 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝑌 − (𝑁‘𝑋))) |
24 | 1, 2, 5, 4, 8, 11, 13, 16, 23 | tgcgrcomlr 27698 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) − 𝐵) = ((𝑁‘𝑋) − 𝑌)) |
25 | 1, 2, 5, 3, 4, 10, 8, 12, 15, 13, 17, 18 | cgr3simp2 27739 | . 2 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝑌 − 𝑍)) |
26 | 1, 2, 5, 6, 7, 4, 8, 9, 10 | mirbtwn 27876 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑀‘𝐴)𝐼𝐴)) |
27 | 1, 6, 5, 4, 11, 10, 8, 26 | btwncolg1 27773 | . . . . 5 ⊢ (𝜑 → (𝐵 ∈ ((𝑀‘𝐴)𝐿𝐴) ∨ (𝑀‘𝐴) = 𝐴)) |
28 | 1, 6, 5, 4, 11, 10, 8, 27 | colcom 27776 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴))) |
29 | 1, 2, 5, 6, 7, 4, 3, 9, 14, 10, 8, 15, 13, 19 | mircgrextend 27900 | . . . . . 6 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋))) |
30 | 1, 2, 5, 4, 10, 11, 15, 16, 29 | tgcgrcomlr 27698 | . . . . 5 ⊢ (𝜑 → ((𝑀‘𝐴) − 𝐴) = ((𝑁‘𝑋) − 𝑋)) |
31 | 1, 2, 3, 4, 10, 8, 11, 15, 13, 16, 19, 23, 30 | trgcgr 27734 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵(𝑀‘𝐴)”〉 ∼ 〈“𝑋𝑌(𝑁‘𝑋)”〉) |
32 | 1, 2, 5, 3, 4, 10, 8, 12, 15, 13, 17, 18 | cgr3simp3 27740 | . . . . 5 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝑍 − 𝑋)) |
33 | 1, 2, 5, 4, 12, 10, 17, 15, 32 | tgcgrcomlr 27698 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝑋 − 𝑍)) |
34 | mirtrcgr.1 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
35 | 1, 6, 5, 4, 10, 8, 11, 3, 15, 13, 2, 12, 16, 17, 28, 31, 33, 25, 34 | tgfscgr 27786 | . . 3 ⊢ (𝜑 → ((𝑀‘𝐴) − 𝐶) = ((𝑁‘𝑋) − 𝑍)) |
36 | 1, 2, 5, 4, 11, 12, 16, 17, 35 | tgcgrcomlr 27698 | . 2 ⊢ (𝜑 → (𝐶 − (𝑀‘𝐴)) = (𝑍 − (𝑁‘𝑋))) |
37 | 1, 2, 3, 4, 11, 8, 12, 16, 13, 17, 24, 25, 36 | trgcgr 27734 | 1 ⊢ (𝜑 → 〈“(𝑀‘𝐴)𝐵𝐶”〉 ∼ 〈“(𝑁‘𝑋)𝑌𝑍”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5144 ‘cfv 6535 (class class class)co 7396 〈“cs3 14780 Basecbs 17131 distcds 17193 TarskiGcstrkg 27645 Itvcitv 27651 LineGclng 27652 cgrGccgrg 27728 pInvGcmir 27870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-oadd 8457 df-er 8691 df-pm 8811 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-dju 9883 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-n0 12460 df-xnn0 12532 df-z 12546 df-uz 12810 df-fz 13472 df-fzo 13615 df-hash 14278 df-word 14452 df-concat 14508 df-s1 14533 df-s2 14786 df-s3 14787 df-trkgc 27666 df-trkgb 27667 df-trkgcb 27668 df-trkg 27671 df-cgrg 27729 df-mir 27871 |
This theorem is referenced by: sacgr 28049 |
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