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Theorem iseqlg 28107

Description: Property of a triangle being equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)

**Hypotheses**
Ref	Expression
iseqlg.p	⊢ 𝑃 = (Base‘𝐺)
iseqlg.m	⊢ − = (dist‘𝐺)
iseqlg.i	⊢ 𝐼 = (Itv‘𝐺)
iseqlg.l	⊢ 𝐿 = (LineG‘𝐺)
iseqlg.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
iseqlg.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
iseqlg.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
iseqlg.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)

**Assertion**
Ref	Expression
iseqlg	⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺) ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐵𝐶𝐴”⟩))

Proof of Theorem iseqlg Dummy variables 𝑥 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqlg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
2		elex 3492	. . . 4 ⊢ (𝐺 ∈ TarskiG → 𝐺 ∈ V)
3		fveq2 6888	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		iseqlg.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
5	3, 4	eqtr4di 2790	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
6	5	oveq1d 7420	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ↑_m (0..^3)) = (𝑃 ↑_m (0..^3)))
7		fveq2 6888	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (cgrG‘𝑔) = (cgrG‘𝐺))
8	7	breqd 5158	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩ ↔ 𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩))
9	6, 8	rabeqbidv 3449	. . . . 5 ⊢ (𝑔 = 𝐺 → {𝑥 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩} = {𝑥 ∈ (𝑃 ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
10		df-eqlg 28106	. . . . 5 ⊢ eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
11		ovex 7438	. . . . . 6 ⊢ (𝑃 ↑_m (0..^3)) ∈ V
12	11	rabex 5331	. . . . 5 ⊢ {𝑥 ∈ (𝑃 ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩} ∈ V
13	9, 10, 12	fvmpt 6995	. . . 4 ⊢ (𝐺 ∈ V → (eqltrG‘𝐺) = {𝑥 ∈ (𝑃 ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
14	1, 2, 13	3syl 18	. . 3 ⊢ (𝜑 → (eqltrG‘𝐺) = {𝑥 ∈ (𝑃 ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
15	14	eleq2d 2819	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ {𝑥 ∈ (𝑃 ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩}))
16		id 22	. . . . 5 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → 𝑥 = ⟨“𝐴𝐵𝐶”⟩)
17		fveq1 6887	. . . . . 6 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → (𝑥‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
18		fveq1 6887	. . . . . 6 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → (𝑥‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
19		fveq1 6887	. . . . . 6 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → (𝑥‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
20	17, 18, 19	s3eqd 14811	. . . . 5 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → ⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩ = ⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩)
21	16, 20	breq12d 5160	. . . 4 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → (𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩))
22	21	elrab 3682	. . 3 ⊢ (⟨“𝐴𝐵𝐶”⟩ ∈ {𝑥 ∈ (𝑃 ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩} ↔ (⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩))
23	22	a1i 11	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ {𝑥 ∈ (𝑃 ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩} ↔ (⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩)))
24		iseqlg.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
25		iseqlg.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
26		iseqlg.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
27	24, 25, 26	s3cld 14819	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
28		s3len 14841	. . . . 5 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
29	4	fvexi 6902	. . . . . 6 ⊢ 𝑃 ∈ V
30		3nn0 12486	. . . . . 6 ⊢ 3 ∈ ℕ₀
31		wrdmap 14492	. . . . . 6 ⊢ ((𝑃 ∈ V ∧ 3 ∈ ℕ₀) → ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))))
32	29, 30, 31	mp2an 690	. . . . 5 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)))
33	27, 28, 32	sylanblc 589	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)))
34	33	biantrurd 533	. . 3 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩ ↔ (⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩)))
35		s3fv1 14839	. . . . . 6 ⊢ (𝐵 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
36	25, 35	syl 17	. . . . 5 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
37		s3fv2 14840	. . . . . 6 ⊢ (𝐶 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
38	26, 37	syl 17	. . . . 5 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
39		s3fv0 14838	. . . . . 6 ⊢ (𝐴 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
40	24, 39	syl 17	. . . . 5 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
41	36, 38, 40	s3eqd 14811	. . . 4 ⊢ (𝜑 → ⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩ = ⟨“𝐵𝐶𝐴”⟩)
42	41	breq2d 5159	. . 3 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐵𝐶𝐴”⟩))
43	34, 42	bitr3d 280	. 2 ⊢ (𝜑 → ((⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)) ∧ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩) ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐵𝐶𝐴”⟩))
44	15, 23, 43	3bitrd 304	1 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺) ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐵𝐶𝐴”⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 0cc0 11106 1c1 11107 2c2 12263 3c3 12264 ℕ₀cn0 12468 ..^cfzo 13623 ♯chash 14286 Word cword 14460 ⟨“cs3 14789 Basecbs 17140 distcds 17202 TarskiGcstrkg 27667 Itvcitv 27673 LineGclng 27674 cgrGccgrg 27750 eqltrGceqlg 28105

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-s3 14796 df-eqlg 28106

This theorem is referenced by: iseqlgd 28108

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