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Theorem hypcgr 28785

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.)

**Hypotheses**
Ref	Expression
hypcgr.p	⊢ 𝑃 = (Base‘𝐺)
hypcgr.m	⊢ − = (dist‘𝐺)
hypcgr.i	⊢ 𝐼 = (Itv‘𝐺)
hypcgr.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
hypcgr.h	⊢ (𝜑 → 𝐺Dim_TarskiG≥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))

**Assertion**
Ref	Expression
hypcgr	⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

**Proof of Theorem hypcgr**
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 → 𝐺Dim_TarskiG≥2)
6		hypcgr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7		hypcgr.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8		hypcgr.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
9		eqid 2736	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
10		eqid 2736	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
11		hypcgr.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
12	1, 2, 3, 4, 5, 7, 11	midcl 28761	. . . 4 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐸) ∈ 𝑃)
13		eqid 2736	. . . 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 28645	. . 3 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) ∈ 𝑃)
16	1, 2, 3, 9, 10, 4, 12, 13, 11	mircl 28645	. . 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 28645	. . 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 28685	. . 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 28654	. . . 4 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸)) = (𝐷 − 𝐸))
24	22, 23	eqtr4d 2774	. . 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 28654	. . . 4 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)) = (𝐸 − 𝐹))
27	25, 26	eqtr4d 2774	. . 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 28766	. . . 4 ⊢ (𝜑 → (𝐸(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐸))
29	1, 2, 3, 4, 5, 11, 7, 10, 12	ismidb 28762	. . . 4 ⊢ (𝜑 → (𝐵 = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸) ↔ (𝐸(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐸)))
30	28, 29	mpbird 257	. . 3 ⊢ (𝜑 → 𝐵 = (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸))
31		eqid 2736	. . 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 28784	. 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 28654	. 2 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)) = (𝐷 − 𝐹))
34	32, 33	eqtrd 2771	1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 2c2 12300 ⟨“cs3 14866 Basecbs 17233 distcds 17285 TarskiGcstrkg 28411 Dim_TarskiG≥cstrkgld 28415 Itvcitv 28417 LineGclng 28418 pInvGcmir 28636 ∟Gcrag 28677 midGcmid 28756 lInvGclmi 28757

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-er 8724 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9920 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-xnn0 12580 df-z 12594 df-uz 12858 df-fz 13530 df-fzo 13677 df-hash 14354 df-word 14537 df-concat 14594 df-s1 14619 df-s2 14872 df-s3 14873 df-trkgc 28432 df-trkgb 28433 df-trkgcb 28434 df-trkgld 28436 df-trkg 28437 df-cgrg 28495 df-ismt 28517 df-leg 28567 df-mir 28637 df-rag 28678 df-perpg 28680 df-mid 28758 df-lmi 28759

This theorem is referenced by: trgcopy 28788

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