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Theorem iseqlg 28622

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 3487	. . . 4 ⊢ (𝐺 ∈ TarskiG → 𝐺 ∈ V)
3		fveq2 6884	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		iseqlg.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
5	3, 4	eqtr4di 2784	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
6	5	oveq1d 7419	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ↑_m (0..^3)) = (𝑃 ↑_m (0..^3)))
7		fveq2 6884	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (cgrG‘𝑔) = (cgrG‘𝐺))
8	7	breqd 5152	. . . . . 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 3443	. . . . 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 28621	. . . . 5 ⊢ eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝑔)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩})
11		ovex 7437	. . . . . 6 ⊢ (𝑃 ↑_m (0..^3)) ∈ V
12	11	rabex 5325	. . . . 5 ⊢ {𝑥 ∈ (𝑃 ↑_m (0..^3)) ∣ 𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩} ∈ V
13	9, 10, 12	fvmpt 6991	. . . 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 2813	. 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 6883	. . . . . 6 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → (𝑥‘1) = (⟨“𝐴𝐵𝐶”⟩‘1))
18		fveq1 6883	. . . . . 6 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → (𝑥‘2) = (⟨“𝐴𝐵𝐶”⟩‘2))
19		fveq1 6883	. . . . . 6 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → (𝑥‘0) = (⟨“𝐴𝐵𝐶”⟩‘0))
20	17, 18, 19	s3eqd 14819	. . . . 5 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → ⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩ = ⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩)
21	16, 20	breq12d 5154	. . . 4 ⊢ (𝑥 = ⟨“𝐴𝐵𝐶”⟩ → (𝑥(cgrG‘𝐺)⟨“(𝑥‘1)(𝑥‘2)(𝑥‘0)”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩))
22	21	elrab 3678	. . 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 14827	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃)
28		s3len 14849	. . . . 5 ⊢ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3
29	4	fvexi 6898	. . . . . 6 ⊢ 𝑃 ∈ V
30		3nn0 12491	. . . . . 6 ⊢ 3 ∈ ℕ₀
31		wrdmap 14500	. . . . . 6 ⊢ ((𝑃 ∈ V ∧ 3 ∈ ℕ₀) → ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3))))
32	29, 30, 31	mp2an 689	. . . . 5 ⊢ ((⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑃 ∧ (♯‘⟨“𝐴𝐵𝐶”⟩) = 3) ↔ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)))
33	27, 28, 32	sylanblc 588	. . . 4 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (𝑃 ↑_m (0..^3)))
34	33	biantrurd 532	. . 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 14847	. . . . . 6 ⊢ (𝐵 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
36	25, 35	syl 17	. . . . 5 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)
37		s3fv2 14848	. . . . . 6 ⊢ (𝐶 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
38	26, 37	syl 17	. . . . 5 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)
39		s3fv0 14846	. . . . . 6 ⊢ (𝐴 ∈ 𝑃 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
40	24, 39	syl 17	. . . . 5 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)
41	36, 38, 40	s3eqd 14819	. . . 4 ⊢ (𝜑 → ⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩ = ⟨“𝐵𝐶𝐴”⟩)
42	41	breq2d 5153	. . 3 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“(⟨“𝐴𝐵𝐶”⟩‘1)(⟨“𝐴𝐵𝐶”⟩‘2)(⟨“𝐴𝐵𝐶”⟩‘0)”⟩ ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐵𝐶𝐴”⟩))
43	34, 42	bitr3d 281	. 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 305	1 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ∈ (eqltrG‘𝐺) ↔ ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐵𝐶𝐴”⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 ↑_m cmap 8819 0cc0 11109 1c1 11110 2c2 12268 3c3 12269 ℕ₀cn0 12473 ..^cfzo 13630 ♯chash 14293 Word cword 14468 ⟨“cs3 14797 Basecbs 17151 distcds 17213 TarskiGcstrkg 28182 Itvcitv 28188 LineGclng 28189 cgrGccgrg 28265 eqltrGceqlg 28620

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-hash 14294 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-eqlg 28621

This theorem is referenced by: iseqlgd 28623

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