![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cgraswaplr | Structured version Visualization version GIF version |
Description: Swap both side of angle congruence. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
Ref | Expression |
---|---|
cgracol.p | ⊢ 𝑃 = (Base‘𝐺) |
cgracol.i | ⊢ 𝐼 = (Itv‘𝐺) |
cgracol.m | ⊢ − = (dist‘𝐺) |
cgracol.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
cgracol.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
cgracol.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
cgracol.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
cgracol.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
cgracol.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
cgracol.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
cgracol.1 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
Ref | Expression |
---|---|
cgraswaplr | ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉(cgrA‘𝐺)〈“𝐹𝐸𝐷”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cgracol.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | cgracol.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | cgracol.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
4 | eqid 2825 | . 2 ⊢ (hlG‘𝐺) = (hlG‘𝐺) | |
5 | cgracol.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
6 | cgracol.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | cgracol.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | cgracol.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | cgracol.e | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
10 | cgracol.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
11 | cgracol.1 | . . . . . 6 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) | |
12 | 1, 2, 4, 3, 7, 6, 5, 8, 9, 10, 11 | cgrane2 26122 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
13 | 12 | necomd 3054 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
14 | 1, 2, 4, 3, 7, 6, 5, 8, 9, 10, 11 | cgrane1 26121 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
15 | 14 | necomd 3054 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
16 | 1, 2, 3, 4, 5, 6, 7, 13, 15 | cgraswap 26129 | . . 3 ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉(cgrA‘𝐺)〈“𝐴𝐵𝐶”〉) |
17 | 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 16, 8, 9, 10, 11 | cgratr 26132 | . 2 ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
18 | 1, 2, 4, 3, 7, 6, 5, 8, 9, 10, 11 | cgrane3 26123 | . . . 4 ⊢ (𝜑 → 𝐸 ≠ 𝐷) |
19 | 18 | necomd 3054 | . . 3 ⊢ (𝜑 → 𝐷 ≠ 𝐸) |
20 | 1, 2, 4, 3, 7, 6, 5, 8, 9, 10, 11 | cgrane4 26124 | . . 3 ⊢ (𝜑 → 𝐸 ≠ 𝐹) |
21 | 1, 2, 3, 4, 8, 9, 10, 19, 20 | cgraswap 26129 | . 2 ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉(cgrA‘𝐺)〈“𝐹𝐸𝐷”〉) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, 10, 9, 8, 21 | cgratr 26132 | 1 ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉(cgrA‘𝐺)〈“𝐹𝐸𝐷”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 class class class wbr 4873 ‘cfv 6123 〈“cs3 13963 Basecbs 16222 distcds 16314 TarskiGcstrkg 25742 Itvcitv 25748 hlGchlg 25912 cgrAccgra 26116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-xnn0 11691 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-hash 13411 df-word 13575 df-concat 13631 df-s1 13656 df-s2 13969 df-s3 13970 df-trkgc 25760 df-trkgb 25761 df-trkgcb 25762 df-trkg 25765 df-cgrg 25823 df-leg 25895 df-hlg 25913 df-cgra 26117 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |