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Theorem hypcgr 28885

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 2737	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
10		eqid 2737	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
11		hypcgr.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
12	1, 2, 3, 4, 5, 7, 11	midcl 28861	. . . 4 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐸) ∈ 𝑃)
13		eqid 2737	. . . 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 28745	. . 3 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) ∈ 𝑃)
16	1, 2, 3, 9, 10, 4, 12, 13, 11	mircl 28745	. . 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 28745	. . 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 28785	. . 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 28754	. . . 4 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸)) = (𝐷 − 𝐸))
24	22, 23	eqtr4d 2775	. . 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 28754	. . . 4 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐸) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)) = (𝐸 − 𝐹))
27	25, 26	eqtr4d 2775	. . 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 28866	. . . 4 ⊢ (𝜑 → (𝐸(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐸))
29	1, 2, 3, 4, 5, 11, 7, 10, 12	ismidb 28862	. . . 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 2737	. . 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 28884	. 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 28754	. 2 ⊢ (𝜑 → ((((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐷) − (((pInvG‘𝐺)‘(𝐵(midG‘𝐺)𝐸))‘𝐹)) = (𝐷 − 𝐹))
34	32, 33	eqtrd 2772	1 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 2c2 12212 ⟨“cs3 14777 Basecbs 17148 distcds 17198 TarskiGcstrkg 28511 Dim_TarskiG≥cstrkgld 28515 Itvcitv 28517 LineGclng 28518 pInvGcmir 28736 ∟Gcrag 28777 midGcmid 28856 lInvGclmi 28857

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-concat 14506 df-s1 14532 df-s2 14783 df-s3 14784 df-trkgc 28532 df-trkgb 28533 df-trkgcb 28534 df-trkgld 28536 df-trkg 28537 df-cgrg 28595 df-ismt 28617 df-leg 28667 df-mir 28737 df-rag 28778 df-perpg 28780 df-mid 28858 df-lmi 28859

This theorem is referenced by: trgcopy 28888

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