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Mirrors > Home > MPE Home > Th. List > hypcgr | Structured version Visualization version GIF version |
Description: If the catheti of two right-angled triangles are congruent, so is their hypothenuse. Theorem 10.12 of [Schwabhauser] p. 91. (Contributed by Thierry Arnoux, 16-Dec-2019.) |
Ref | Expression |
---|---|
hypcgr.p | ⊢ 𝑃 = (Base‘𝐺) |
hypcgr.m | ⊢ − = (dist‘𝐺) |
hypcgr.i | ⊢ 𝐼 = (Itv‘𝐺) |
hypcgr.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
hypcgr.h | ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
hypcgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
hypcgr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
hypcgr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
hypcgr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
hypcgr.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
hypcgr.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
hypcgr.1 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
hypcgr.2 | ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉 ∈ (∟G‘𝐺)) |
hypcgr.3 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) |
hypcgr.4 | ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) |
Ref | Expression |
---|---|
hypcgr | ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hypcgr.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | hypcgr.m | . . 3 ⊢ − = (dist‘𝐺) | |
3 | hypcgr.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hypcgr.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | hypcgr.h | . . 3 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
6 | hypcgr.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | hypcgr.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | hypcgr.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | eqid 2778 | . . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺) | |
10 | eqid 2778 | . . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺) | |
11 | hypcgr.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
12 | 1, 2, 3, 4, 5, 7, 11 | midcl 26142 | . . . 4 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐸) ∈ 𝑃) |
13 | eqid 2778 | . . . 4 ⊢ ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸)) = ((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸)) | |
14 | hypcgr.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
15 | 1, 2, 3, 9, 10, 4, 12, 13, 14 | mircl 26029 | . . 3 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) ∈ 𝑃) |
16 | 1, 2, 3, 9, 10, 4, 12, 13, 11 | mircl 26029 | . . 3 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸) ∈ 𝑃) |
17 | hypcgr.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
18 | 1, 2, 3, 9, 10, 4, 12, 13, 17 | mircl 26029 | . . 3 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹) ∈ 𝑃) |
19 | hypcgr.1 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) | |
20 | hypcgr.2 | . . . 4 ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉 ∈ (∟G‘𝐺)) | |
21 | 1, 2, 3, 9, 10, 4, 14, 11, 17, 20, 13, 12 | mirrag 26069 | . . 3 ⊢ (𝜑 → 〈“(((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷)(((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸)(((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)”〉 ∈ (∟G‘𝐺)) |
22 | hypcgr.3 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) | |
23 | 1, 2, 3, 9, 10, 4, 12, 13, 14, 11 | miriso 26038 | . . . 4 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸)) = (𝐷 − 𝐸)) |
24 | 22, 23 | eqtr4d 2817 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸))) |
25 | hypcgr.4 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) | |
26 | 1, 2, 3, 9, 10, 4, 12, 13, 11, 17 | miriso 26038 | . . . 4 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)) = (𝐸 − 𝐹)) |
27 | 25, 26 | eqtr4d 2817 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐶) = ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹))) |
28 | 1, 2, 3, 4, 5, 11, 7 | midcom 26147 | . . . 4 ⊢ (𝜑 → (𝐸(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐸)) |
29 | 1, 2, 3, 4, 5, 11, 7, 10, 12 | ismidb 26143 | . . . 4 ⊢ (𝜑 → (𝐵 = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸) ↔ (𝐸(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐸))) |
30 | 28, 29 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝐵 = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸)) |
31 | eqid 2778 | . . 3 ⊢ ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)(((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹))(LineG‘𝐺)𝐵)) = ((lInvG‘𝐺)‘((𝐶(midG‘𝐺)(((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹))(LineG‘𝐺)𝐵)) | |
32 | 1, 2, 3, 4, 5, 6, 7, 8, 15, 16, 18, 19, 21, 24, 27, 30, 31 | hypcgrlem2 26165 | . 2 ⊢ (𝜑 → (𝐴 − 𝐶) = ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹))) |
33 | 1, 2, 3, 9, 10, 4, 12, 13, 14, 17 | miriso 26038 | . 2 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)) = (𝐷 − 𝐹)) |
34 | 32, 33 | eqtrd 2814 | 1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 2c2 11435 〈“cs3 13999 Basecbs 16266 distcds 16358 TarskiGcstrkg 25798 DimTarskiG≥cstrkgld 25802 Itvcitv 25804 LineGclng 25805 pInvGcmir 26020 ∟Gcrag 26061 midGcmid 26137 lInvGclmi 26138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-xnn0 11720 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-hash 13442 df-word 13606 df-concat 13667 df-s1 13692 df-s2 14005 df-s3 14006 df-trkgc 25816 df-trkgb 25817 df-trkgcb 25818 df-trkgld 25820 df-trkg 25821 df-cgrg 25879 df-ismt 25901 df-leg 25951 df-mir 26021 df-rag 26062 df-perpg 26064 df-mid 26139 df-lmi 26140 |
This theorem is referenced by: trgcopy 26169 |
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