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Mirrors > Home > MPE Home > Th. List > cgr3swap23 | Structured version Visualization version GIF version |
Description: Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019.) |
Ref | Expression |
---|---|
tgcgrxfr.p | ⊢ 𝑃 = (Base‘𝐺) |
tgcgrxfr.m | ⊢ − = (dist‘𝐺) |
tgcgrxfr.i | ⊢ 𝐼 = (Itv‘𝐺) |
tgcgrxfr.r | ⊢ ∼ = (cgrG‘𝐺) |
tgcgrxfr.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwnxfr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwnxfr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgbtwnxfr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgbtwnxfr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tgbtwnxfr.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
tgbtwnxfr.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
tgbtwnxfr.2 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) |
Ref | Expression |
---|---|
cgr3swap23 | ⊢ (𝜑 → 〈“𝐴𝐶𝐵”〉 ∼ 〈“𝐷𝐹𝐸”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcgrxfr.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tgcgrxfr.m | . 2 ⊢ − = (dist‘𝐺) | |
3 | tgcgrxfr.r | . 2 ⊢ ∼ = (cgrG‘𝐺) | |
4 | tgcgrxfr.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tgbtwnxfr.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | tgbtwnxfr.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | tgbtwnxfr.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | tgbtwnxfr.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | tgbtwnxfr.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
10 | tgbtwnxfr.e | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
11 | tgcgrxfr.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
12 | tgbtwnxfr.2 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) | |
13 | 1, 2, 11, 3, 4, 5, 7, 6, 8, 10, 9, 12 | cgr3simp3 26022 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
14 | 1, 2, 11, 4, 6, 5, 9, 8, 13 | tgcgrcomlr 25980 | . 2 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
15 | 1, 2, 11, 3, 4, 5, 7, 6, 8, 10, 9, 12 | cgr3simp2 26021 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹)) |
16 | 1, 2, 11, 4, 7, 6, 10, 9, 15 | tgcgrcomlr 25980 | . 2 ⊢ (𝜑 → (𝐶 − 𝐵) = (𝐹 − 𝐸)) |
17 | 1, 2, 11, 3, 4, 5, 7, 6, 8, 10, 9, 12 | cgr3simp1 26020 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸)) |
18 | 1, 2, 11, 4, 5, 7, 8, 10, 17 | tgcgrcomlr 25980 | . 2 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷)) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 18 | trgcgr 26016 | 1 ⊢ (𝜑 → 〈“𝐴𝐶𝐵”〉 ∼ 〈“𝐷𝐹𝐸”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 class class class wbr 4925 ‘cfv 6185 〈“cs3 14064 Basecbs 16337 distcds 16428 TarskiGcstrkg 25930 Itvcitv 25936 cgrGccgrg 26010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-pm 8207 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 df-fzo 12848 df-hash 13504 df-word 13671 df-concat 13732 df-s1 13757 df-s2 14070 df-s3 14071 df-trkgc 25948 df-trkgcb 25950 df-trkg 25953 df-cgrg 26011 |
This theorem is referenced by: cgr3swap13 26025 cgr3rotr 26026 cgr3rotl 26027 lnxfr 26066 lnext 26067 tgfscgr 26068 legov 26085 legov2 26086 legtrd 26089 symquadlem 26189 |
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