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Mirrors > Home > MPE Home > Th. List > tgsas | 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 the triangles are congruent. 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 |
---|---|
tgsas | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgsas.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tgsas.m | . 2 ⊢ − = (dist‘𝐺) | |
3 | eqid 2778 | . 2 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺) | |
4 | tgsas.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tgsas.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | tgsas.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | tgsas.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
8 | tgsas.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | tgsas.e | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
10 | tgsas.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
11 | tgsas.1 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) | |
12 | tgsas.3 | . 2 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) | |
13 | tgsas.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
14 | tgsas.2 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) | |
15 | 1, 2, 13, 4, 5, 6, 7, 8, 9, 10, 11, 14, 12 | tgsas1 26220 | . 2 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15 | trgcgr 25884 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 〈“cs3 13999 Basecbs 16266 distcds 16358 TarskiGcstrkg 25798 Itvcitv 25804 cgrGccgrg 25878 cgrAccgra 26172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-xnn0 11720 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-hash 13442 df-word 13606 df-concat 13667 df-s1 13692 df-s2 14005 df-s3 14006 df-trkgc 25816 df-trkgb 25817 df-trkgcb 25818 df-trkg 25821 df-cgrg 25879 df-leg 25951 df-hlg 25969 df-cgra 26173 |
This theorem is referenced by: tgsas2 26222 tgsas3 26223 tgasa 26225 |
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