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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cgr3permute3 | Structured version Visualization version GIF version |
Description: Permutation law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.) |
Ref | Expression |
---|---|
cgr3permute3 | ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 〈𝐵, 〈𝐶, 𝐴〉〉Cgr3〈𝐸, 〈𝐹, 𝐷〉〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ) | |
2 | 3simpa 1141 | . . . . 5 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) | |
3 | 3simpa 1141 | . . . . 5 ⊢ ((𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)) → (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) | |
4 | cgrcomlr 33069 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → (〈𝐴, 𝐵〉Cgr〈𝐷, 𝐸〉 ↔ 〈𝐵, 𝐴〉Cgr〈𝐸, 𝐷〉)) | |
5 | 1, 2, 3, 4 | syl3an 1153 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 𝐵〉Cgr〈𝐷, 𝐸〉 ↔ 〈𝐵, 𝐴〉Cgr〈𝐸, 𝐷〉)) |
6 | 3simpb 1142 | . . . . 5 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) | |
7 | 3simpb 1142 | . . . . 5 ⊢ ((𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)) → (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) | |
8 | cgrcomlr 33069 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 𝐶〉Cgr〈𝐷, 𝐹〉 ↔ 〈𝐶, 𝐴〉Cgr〈𝐹, 𝐷〉)) | |
9 | 1, 6, 7, 8 | syl3an 1153 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 𝐶〉Cgr〈𝐷, 𝐹〉 ↔ 〈𝐶, 𝐴〉Cgr〈𝐹, 𝐷〉)) |
10 | 5, 9 | 3anbi12d 1429 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((〈𝐴, 𝐵〉Cgr〈𝐷, 𝐸〉 ∧ 〈𝐴, 𝐶〉Cgr〈𝐷, 𝐹〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉) ↔ (〈𝐵, 𝐴〉Cgr〈𝐸, 𝐷〉 ∧ 〈𝐶, 𝐴〉Cgr〈𝐹, 𝐷〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉))) |
11 | 3anrot 1093 | . . 3 ⊢ ((〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉 ∧ 〈𝐵, 𝐴〉Cgr〈𝐸, 𝐷〉 ∧ 〈𝐶, 𝐴〉Cgr〈𝐹, 𝐷〉) ↔ (〈𝐵, 𝐴〉Cgr〈𝐸, 𝐷〉 ∧ 〈𝐶, 𝐴〉Cgr〈𝐹, 𝐷〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉)) | |
12 | 10, 11 | syl6bbr 290 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((〈𝐴, 𝐵〉Cgr〈𝐷, 𝐸〉 ∧ 〈𝐴, 𝐶〉Cgr〈𝐷, 𝐹〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉) ↔ (〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉 ∧ 〈𝐵, 𝐴〉Cgr〈𝐸, 𝐷〉 ∧ 〈𝐶, 𝐴〉Cgr〈𝐹, 𝐷〉))) |
13 | brcgr3 33117 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ↔ (〈𝐴, 𝐵〉Cgr〈𝐷, 𝐸〉 ∧ 〈𝐴, 𝐶〉Cgr〈𝐷, 𝐹〉 ∧ 〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉))) | |
14 | biid 262 | . . 3 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ ℕ) | |
15 | 3anrot 1093 | . . 3 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) | |
16 | 3anrot 1093 | . . 3 ⊢ ((𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)) ↔ (𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) | |
17 | brcgr3 33117 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (〈𝐵, 〈𝐶, 𝐴〉〉Cgr3〈𝐸, 〈𝐹, 𝐷〉〉 ↔ (〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉 ∧ 〈𝐵, 𝐴〉Cgr〈𝐸, 𝐷〉 ∧ 〈𝐶, 𝐴〉Cgr〈𝐹, 𝐷〉))) | |
18 | 14, 15, 16, 17 | syl3anb 1154 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐵, 〈𝐶, 𝐴〉〉Cgr3〈𝐸, 〈𝐹, 𝐷〉〉 ↔ (〈𝐵, 𝐶〉Cgr〈𝐸, 𝐹〉 ∧ 〈𝐵, 𝐴〉Cgr〈𝐸, 𝐷〉 ∧ 〈𝐶, 𝐴〉Cgr〈𝐹, 𝐷〉))) |
19 | 12, 13, 18 | 3bitr4d 312 | 1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (〈𝐴, 〈𝐵, 𝐶〉〉Cgr3〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 〈𝐵, 〈𝐶, 𝐴〉〉Cgr3〈𝐸, 〈𝐹, 𝐷〉〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 ∈ wcel 2081 〈cop 4478 class class class wbr 4962 ‘cfv 6225 ℕcn 11486 𝔼cee 26357 Cgrccgr 26359 Cgr3ccgr3 33107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 df-seq 13220 df-exp 13280 df-sum 14877 df-ee 26360 df-cgr 26362 df-cgr3 33112 |
This theorem is referenced by: cgr3permute2 33120 cgr3permute4 33121 cgr3permute5 33122 colinearxfr 33146 |
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