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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 2622 | . 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 25705 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
13 | 12 | necomd 2849 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
14 | 1, 2, 4, 3, 7, 6, 5, 8, 9, 10, 11 | cgrane1 25704 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
15 | 14 | necomd 2849 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
16 | 1, 2, 3, 4, 5, 6, 7, 13, 15 | cgraswap 25712 | . . 3 ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉(cgrA‘𝐺)〈“𝐴𝐵𝐶”〉) |
17 | 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 16, 8, 9, 10, 11 | cgratr 25715 | . 2 ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
18 | 1, 2, 4, 3, 7, 6, 5, 8, 9, 10, 11 | cgrane3 25706 | . . . 4 ⊢ (𝜑 → 𝐸 ≠ 𝐷) |
19 | 18 | necomd 2849 | . . 3 ⊢ (𝜑 → 𝐷 ≠ 𝐸) |
20 | 1, 2, 4, 3, 7, 6, 5, 8, 9, 10, 11 | cgrane4 25707 | . . 3 ⊢ (𝜑 → 𝐸 ≠ 𝐹) |
21 | 1, 2, 3, 4, 8, 9, 10, 19, 20 | cgraswap 25712 | . 2 ⊢ (𝜑 → 〈“𝐷𝐸𝐹”〉(cgrA‘𝐺)〈“𝐹𝐸𝐷”〉) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, 10, 9, 8, 21 | cgratr 25715 | 1 ⊢ (𝜑 → 〈“𝐶𝐵𝐴”〉(cgrA‘𝐺)〈“𝐹𝐸𝐷”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 〈“cs3 13587 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 hlGchlg 25495 cgrAccgra 25699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-cgrg 25406 df-leg 25478 df-hlg 25496 df-cgra 25700 |
This theorem is referenced by: (None) |
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