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Mirrors > Home > MPE Home > Th. List > tgsas3 | Structured version Visualization version GIF version |
Description: First congruence theorem: SAS. 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)) |
tgsas2.4 | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Ref | Expression |
---|---|
tgsas3 | ⊢ (𝜑 → 〈“𝐵𝐶𝐴”〉(cgrA‘𝐺)〈“𝐸𝐹𝐷”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgsas.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tgsas.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | eqid 2821 | . 2 ⊢ (hlG‘𝐺) = (hlG‘𝐺) | |
4 | tgsas.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tgsas.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
6 | tgsas.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | tgsas.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | tgsas.e | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
9 | tgsas.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
10 | tgsas.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
11 | tgsas.m | . . 3 ⊢ − = (dist‘𝐺) | |
12 | eqid 2821 | . . 3 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺) | |
13 | tgsas.1 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) | |
14 | tgsas.2 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) | |
15 | tgsas.3 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) | |
16 | 1, 11, 2, 4, 7, 5, 6, 10, 8, 9, 13, 14, 15 | tgsas 26640 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹”〉) |
17 | 1, 11, 2, 12, 4, 7, 5, 6, 10, 8, 9, 16 | cgr3rotl 26312 | . 2 ⊢ (𝜑 → 〈“𝐵𝐶𝐴”〉(cgrG‘𝐺)〈“𝐸𝐹𝐷”〉) |
18 | 1, 2, 3, 4, 7, 5, 6, 10, 8, 9, 14 | cgrane4 26600 | . . 3 ⊢ (𝜑 → 𝐸 ≠ 𝐹) |
19 | 1, 2, 3, 8, 7, 9, 4, 18 | hlid 26394 | . 2 ⊢ (𝜑 → 𝐸((hlG‘𝐺)‘𝐹)𝐸) |
20 | 1, 11, 2, 4, 7, 5, 6, 10, 8, 9, 13, 14, 15 | tgsas1 26639 | . . . . 5 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
21 | 1, 11, 2, 4, 6, 7, 9, 10, 20 | tgcgrcomlr 26265 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
22 | tgsas2.4 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
23 | 1, 11, 2, 4, 7, 6, 10, 9, 21, 22 | tgcgrneq 26268 | . . 3 ⊢ (𝜑 → 𝐷 ≠ 𝐹) |
24 | 1, 2, 3, 10, 7, 9, 4, 23 | hlid 26394 | . 2 ⊢ (𝜑 → 𝐷((hlG‘𝐺)‘𝐹)𝐷) |
25 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 8, 10, 17, 19, 24 | iscgrad 26596 | 1 ⊢ (𝜑 → 〈“𝐵𝐶𝐴”〉(cgrA‘𝐺)〈“𝐸𝐹𝐷”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 〈“cs3 14203 Basecbs 16482 distcds 16573 TarskiGcstrkg 26215 Itvcitv 26221 cgrGccgrg 26295 hlGchlg 26385 cgrAccgra 26592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-s2 14209 df-s3 14210 df-trkgc 26233 df-trkgb 26234 df-trkgcb 26235 df-trkg 26238 df-cgrg 26296 df-leg 26368 df-hlg 26386 df-cgra 26593 |
This theorem is referenced by: (None) |
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