Theorem ltgseg 25947

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 774	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( − ‘𝑎) = 𝐴)
2		simpr 479	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝑎 = ⟨𝑥, 𝑦⟩)
3	2	fveq2d 6450	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( − ‘𝑎) = ( − ‘⟨𝑥, 𝑦⟩))
4	1, 3	eqtr3d 2815	. . . 4 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = ( − ‘⟨𝑥, 𝑦⟩))
5		df-ov 6925	. . . 4 ⊢ (𝑥 − 𝑦) = ( − ‘⟨𝑥, 𝑦⟩)
6	4, 5	syl6eqr 2831	. . 3 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = (𝑥 − 𝑦))
7		simplr 759	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃))
8		elxp2 5379	. . . 4 ⊢ (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
9	7, 8	sylib 210	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
10	6, 9	reximddv2 3201	. 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	syl6eleq 2868	. . 3 ⊢ (𝜑 → 𝐴 ∈ ( − “ (𝑃 × 𝑃)))
15		fvelima 6508	. . 3 ⊢ ((Fun − ∧ 𝐴 ∈ ( − “ (𝑃 × 𝑃))) → ∃𝑎 ∈ (𝑃 × 𝑃)( − ‘𝑎) = 𝐴)
16	11, 14, 15	syl2anc 579	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑃 × 𝑃)( − ‘𝑎) = 𝐴)
17	10, 16	r19.29a 3263	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝐴 = (𝑥 − 𝑦))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106  ∃wrex 3090  ⟨cop 4403   × cxp 5353   “ cima 5358  Fun wfun 6129  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  distcds 16347  TarskiGcstrkg 25781  Itvcitv 25787  ≤Gcleg 25933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925

This theorem is referenced by:  legso  25950