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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 2821 | . . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺) | |
10 | eqid 2821 | . . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺) | |
11 | hypcgr.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
12 | 1, 2, 3, 4, 5, 7, 11 | midcl 26557 | . . . 4 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐸) ∈ 𝑃) |
13 | eqid 2821 | . . . 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 26441 | . . 3 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) ∈ 𝑃) |
16 | 1, 2, 3, 9, 10, 4, 12, 13, 11 | mircl 26441 | . . 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 26441 | . . 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 26481 | . . 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 26450 | . . . 4 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸)) = (𝐷 − 𝐸)) |
24 | 22, 23 | eqtr4d 2859 | . . 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 26450 | . . . 4 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)) = (𝐸 − 𝐹)) |
27 | 25, 26 | eqtr4d 2859 | . . 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 26562 | . . . 4 ⊢ (𝜑 → (𝐸(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐸)) |
29 | 1, 2, 3, 4, 5, 11, 7, 10, 12 | ismidb 26558 | . . . 4 ⊢ (𝜑 → (𝐵 = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸) ↔ (𝐸(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐸))) |
30 | 28, 29 | mpbird 259 | . . 3 ⊢ (𝜑 → 𝐵 = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸)) |
31 | eqid 2821 | . . 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 26580 | . 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 26450 | . 2 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)) = (𝐷 − 𝐹)) |
34 | 32, 33 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 2c2 11686 〈“cs3 14198 Basecbs 16477 distcds 16568 TarskiGcstrkg 26210 DimTarskiG≥cstrkgld 26214 Itvcitv 26216 LineGclng 26217 pInvGcmir 26432 ∟Gcrag 26473 midGcmid 26552 lInvGclmi 26553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-trkgc 26228 df-trkgb 26229 df-trkgcb 26230 df-trkgld 26232 df-trkg 26233 df-cgrg 26291 df-ismt 26313 df-leg 26363 df-mir 26433 df-rag 26474 df-perpg 26476 df-mid 26554 df-lmi 26555 |
This theorem is referenced by: trgcopy 26584 |
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