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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 2726 | . 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 2726 | . . 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 28782 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝐷𝐸𝐹”〉) |
17 | 1, 11, 2, 12, 4, 7, 5, 6, 10, 8, 9, 16 | cgr3rotl 28454 | . 2 ⊢ (𝜑 → 〈“𝐵𝐶𝐴”〉(cgrG‘𝐺)〈“𝐸𝐹𝐷”〉) |
18 | 1, 2, 3, 4, 7, 5, 6, 10, 8, 9, 14 | cgrane4 28742 | . . 3 ⊢ (𝜑 → 𝐸 ≠ 𝐹) |
19 | 1, 2, 3, 8, 7, 9, 4, 18 | hlid 28536 | . 2 ⊢ (𝜑 → 𝐸((hlG‘𝐺)‘𝐹)𝐸) |
20 | 1, 11, 2, 4, 7, 5, 6, 10, 8, 9, 13, 14, 15 | tgsas1 28781 | . . . . 5 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
21 | 1, 11, 2, 4, 6, 7, 9, 10, 20 | tgcgrcomlr 28407 | . . . 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 28410 | . . 3 ⊢ (𝜑 → 𝐷 ≠ 𝐹) |
24 | 1, 2, 3, 10, 7, 9, 4, 23 | hlid 28536 | . 2 ⊢ (𝜑 → 𝐷((hlG‘𝐺)‘𝐹)𝐷) |
25 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 8, 10, 17, 19, 24 | iscgrad 28738 | 1 ⊢ (𝜑 → 〈“𝐵𝐶𝐴”〉(cgrA‘𝐺)〈“𝐸𝐹𝐷”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 〈“cs3 14851 Basecbs 17213 distcds 17275 TarskiGcstrkg 28354 Itvcitv 28360 cgrGccgrg 28437 hlGchlg 28527 cgrAccgra 28734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-oadd 8500 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12597 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-hash 14348 df-word 14523 df-concat 14579 df-s1 14604 df-s2 14857 df-s3 14858 df-trkgc 28375 df-trkgb 28376 df-trkgcb 28377 df-trkg 28380 df-cgrg 28438 df-leg 28510 df-hlg 28528 df-cgra 28735 |
This theorem is referenced by: (None) |
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