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Mirrors > Home > MPE Home > Th. List > tgsas1 | Structured version Visualization version GIF version |
Description: First congruence theorem: SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then third sides are equal in length. Theorem 11.49 of [Schwabhauser] p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020.) |
Ref | Expression |
---|---|
tgsas.p | ⊢ 𝑃 = (Base‘𝐺) |
tgsas.m | ⊢ − = (dist‘𝐺) |
tgsas.i | ⊢ 𝐼 = (Itv‘𝐺) |
tgsas.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgsas.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgsas.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgsas.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgsas.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tgsas.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
tgsas.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
tgsas.1 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) |
tgsas.2 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
tgsas.3 | ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) |
Ref | Expression |
---|---|
tgsas1 | ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgsas.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tgsas.m | . 2 ⊢ − = (dist‘𝐺) | |
3 | tgsas.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tgsas.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tgsas.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | tgsas.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | tgsas.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
8 | tgsas.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
9 | eqid 2736 | . . 3 ⊢ (hlG‘𝐺) = (hlG‘𝐺) | |
10 | tgsas.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | tgsas.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
12 | tgsas.2 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) | |
13 | 1, 3, 9, 4, 5, 10, 6, 7, 11, 8, 12 | cgrane1 26857 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
14 | 1, 3, 9, 5, 5, 10, 4, 13 | hlid 26654 | . . 3 ⊢ (𝜑 → 𝐴((hlG‘𝐺)‘𝐵)𝐴) |
15 | 1, 3, 9, 4, 5, 10, 6, 7, 11, 8, 12 | cgrane2 26858 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
16 | 15 | necomd 2987 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
17 | 1, 3, 9, 6, 5, 10, 4, 16 | hlid 26654 | . . 3 ⊢ (𝜑 → 𝐶((hlG‘𝐺)‘𝐵)𝐶) |
18 | tgsas.1 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) | |
19 | 1, 2, 3, 4, 5, 10, 7, 11, 18 | tgcgrcomlr 26525 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷)) |
20 | tgsas.3 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) | |
21 | 1, 3, 9, 4, 5, 10, 6, 7, 11, 8, 12, 5, 2, 6, 14, 17, 19, 20 | cgracgr 26863 | . 2 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 21 | tgcgrcomlr 26525 | 1 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 〈“cs3 14372 Basecbs 16666 distcds 16758 TarskiGcstrkg 26475 Itvcitv 26481 hlGchlg 26645 cgrAccgra 26852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-xnn0 12128 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-concat 14091 df-s1 14118 df-s2 14378 df-s3 14379 df-trkgc 26493 df-trkgb 26494 df-trkgcb 26495 df-trkg 26498 df-cgrg 26556 df-leg 26628 df-hlg 26646 df-cgra 26853 |
This theorem is referenced by: tgsas 26900 tgsas2 26901 tgsas3 26902 |
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