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Theorem hypcgr 28957

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 2761	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
10		eqid 2761	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
11		hypcgr.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
12	1, 2, 3, 4, 5, 7, 11	midcl 28933	. . . 4 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐸) ∈ 𝑃)
13		eqid 2761	. . . 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 28817	. . 3 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) ∈ 𝑃)
16	1, 2, 3, 9, 10, 4, 12, 13, 11	mircl 28817	. . 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 28817	. . 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 28857	. . 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 28826	. . . 4 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸)) = (𝐷 − 𝐸))
24	22, 23	eqtr4d 2799	. . 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 28826	. . . 4 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)) = (𝐸 − 𝐹))
27	25, 26	eqtr4d 2799	. . 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 28938	. . . 4 ⊢ (𝜑 → (𝐸(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐸))
29	1, 2, 3, 4, 5, 11, 7, 10, 12	ismidb 28934	. . . 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 2761	. . 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 28956	. 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 28826	. 2 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)) = (𝐷 − 𝐹))
34	32, 33	eqtrd 2796	1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 class class class wbr 5097 ‘cfv 6515 (class class class)co 7390 2c2 12265 ⟨“cs3 14848 Basecbs 17235 distcds 17285 TarskiGcstrkg 28583 Dim_TarskiG≥cstrkgld 28587 Itvcitv 28589 LineGclng 28590 pInvGcmir 28808 ∟Gcrag 28849 midGcmid 28928 lInvGclmi 28929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-oadd 8434 df-er 8671 df-map 8803 df-pm 8804 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-dju 9852 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-nn 12204 df-2 12273 df-3 12274 df-n0 12475 df-xnn0 12548 df-z 12562 df-uz 12833 df-fz 13506 df-fzo 13653 df-hash 14337 df-word 14520 df-concat 14577 df-s1 14603 df-s2 14854 df-s3 14855 df-trkgc 28604 df-trkgb 28605 df-trkgcb 28606 df-trkgld 28608 df-trkg 28609 df-cgrg 28667 df-ismt 28689 df-leg 28739 df-mir 28809 df-rag 28850 df-perpg 28852 df-mid 28930 df-lmi 28931

This theorem is referenced by: trgcopy 28960

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