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Mirrors > Home > MPE Home > Th. List > isoas | Structured version Visualization version GIF version |
Description: Congruence theorem for isocele triangles: if two angles of a triangle are congruent, then the corresponding sides also are. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
Ref | Expression |
---|---|
isoas.p | ⊢ 𝑃 = (Base‘𝐺) |
isoas.m | ⊢ − = (dist‘𝐺) |
isoas.i | ⊢ 𝐼 = (Itv‘𝐺) |
isoas.l | ⊢ 𝐿 = (LineG‘𝐺) |
isoas.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
isoas.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
isoas.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
isoas.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
isoas.1 | ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) |
isoas.2 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐴𝐶𝐵”〉) |
Ref | Expression |
---|---|
isoas | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isoas.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | isoas.m | . 2 ⊢ − = (dist‘𝐺) | |
3 | isoas.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | eqid 2824 | . 2 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺) | |
5 | isoas.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | isoas.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | isoas.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
8 | isoas.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | isoas.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
10 | isoas.1 | . . . 4 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) | |
11 | 1, 9, 3, 5, 8, 6, 7, 10 | ncolrot1 26351 | . . 3 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) |
12 | 1, 2, 3, 5, 6, 7 | axtgcgrrflx 26251 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐶 − 𝐵)) |
13 | eqid 2824 | . . . . 5 ⊢ (hlG‘𝐺) = (hlG‘𝐺) | |
14 | isoas.2 | . . . . 5 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐴𝐶𝐵”〉) | |
15 | 1, 3, 5, 13, 8, 6, 7, 8, 7, 6, 14 | cgracom 26611 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐶𝐵”〉(cgrA‘𝐺)〈“𝐴𝐵𝐶”〉) |
16 | 1, 3, 2, 5, 8, 7, 6, 8, 6, 7, 15 | cgraswaplr 26614 | . . 3 ⊢ (𝜑 → 〈“𝐵𝐶𝐴”〉(cgrA‘𝐺)〈“𝐶𝐵𝐴”〉) |
17 | 1, 2, 3, 5, 6, 7, 8, 7, 6, 8, 9, 11, 12, 16, 14 | tgasa 26648 | . 2 ⊢ (𝜑 → 〈“𝐵𝐶𝐴”〉(cgrG‘𝐺)〈“𝐶𝐵𝐴”〉) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 8, 17 | cgr3simp3 26311 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 843 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 〈“cs3 14207 Basecbs 16486 distcds 16577 TarskiGcstrkg 26219 Itvcitv 26225 LineGclng 26226 cgrGccgrg 26299 hlGchlg 26389 cgrAccgra 26596 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13926 df-s1 13953 df-s2 14213 df-s3 14214 df-trkgc 26237 df-trkgb 26238 df-trkgcb 26239 df-trkgld 26241 df-trkg 26242 df-cgrg 26300 df-leg 26372 df-hlg 26390 df-mir 26442 df-rag 26483 df-perpg 26485 df-hpg 26547 df-mid 26563 df-lmi 26564 df-cgra 26597 |
This theorem is referenced by: (None) |
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