Theorem ltgseg 28902

Description: The set 𝐸 denotes the possible values of the congruence. (Contributed by Thierry Arnoux, 15-Dec-2019.)

Hypotheses

Ref	Expression
legval.p	⊢ 𝑃 = (Base‘𝐺)
legval.d	⊢ − = (dist‘𝐺)
legval.i	⊢ 𝐼 = (Itv‘𝐺)
legval.l	⊢ ≤ = (≤G‘𝐺)
legval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
legso.a	⊢ 𝐸 = ( − “ (𝑃 × 𝑃))
legso.f	⊢ (𝜑 → Fun − )
ltgseg.p	⊢ (𝜑 → 𝐴 ∈ 𝐸)

Assertion

Ref	Expression
ltgseg	⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝐴 = (𝑥 − 𝑦))

Distinct variable groups:   𝑥, − ,𝑦   𝑥,𝐴,𝑦   𝑥,𝑃,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐸(𝑥, 𝑦)   𝐺(𝑥, 𝑦)   𝐼(𝑥, 𝑦)   ≤ (𝑥, 𝑦)

Proof of Theorem ltgseg

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp-4r 796	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( − ‘𝑎) = 𝐴)
2		simpr 490	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝑎 = ⟨𝑥, 𝑦⟩)
3	2	fveq2d 6892	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( − ‘𝑎) = ( − ‘⟨𝑥, 𝑦⟩))
4	1, 3	eqtr3d 2803	. . . 4 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = ( − ‘⟨𝑥, 𝑦⟩))
5		df-ov 7426	. . . 4 ⊢ (𝑥 − 𝑦) = ( − ‘⟨𝑥, 𝑦⟩)
6	4, 5	eqtr4di 2819	. . 3 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = (𝑥 − 𝑦))
7		simplr 781	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃))
8		elxp2 5690	. . . 4 ⊢ (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
9	7, 8	sylib 221	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
10	6, 9	reximddv2 3227	. 2 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝐴 = (𝑥 − 𝑦))
11		legso.f	. . 3 ⊢ (𝜑 → Fun − )
12		ltgseg.p	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐸)
13		legso.a	. . . 4 ⊢ 𝐸 = ( − “ (𝑃 × 𝑃))
14	12, 13	eleqtrdi 2876	. . 3 ⊢ (𝜑 → 𝐴 ∈ ( − “ (𝑃 × 𝑃)))
15		fvelima 6953	. . 3 ⊢ ((Fun − ∧ 𝐴 ∈ ( − “ (𝑃 × 𝑃))) → ∃𝑎 ∈ (𝑃 × 𝑃)( − ‘𝑎) = 𝐴)
16	11, 14, 15	syl2anc 596	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑃 × 𝑃)( − ‘𝑎) = 𝐴)
17	10, 16	r19.29a 3176	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝐴 = (𝑥 − 𝑦))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3092  ⟨cop 4600   × cxp 5664   “ cima 5669  Fun wfun 6537  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  distcds 17344  TarskiGcstrkg 28733  Itvcitv 28739  ≤Gcleg 28888

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426

This theorem is used by:  legso  28905