Theorem ltgseg 25491

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 807	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( − ‘𝑎) = 𝐴)
2		simpr 477	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝑎 = ⟨𝑥, 𝑦⟩)
3	2	fveq2d 6195	. . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → ( − ‘𝑎) = ( − ‘⟨𝑥, 𝑦⟩))
4	1, 3	eqtr3d 2658	. . . 4 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = ( − ‘⟨𝑥, 𝑦⟩))
5		df-ov 6653	. . . 4 ⊢ (𝑥 − 𝑦) = ( − ‘⟨𝑥, 𝑦⟩)
6	4, 5	syl6eqr 2674	. . 3 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = ⟨𝑥, 𝑦⟩) → 𝐴 = (𝑥 − 𝑦))
7		simplr 792	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃))
8		elxp2 5132	. . . 4 ⊢ (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
9	7, 8	sylib 208	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = ⟨𝑥, 𝑦⟩)
10	6, 9	reximddv2 3020	. 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 2711	. . 3 ⊢ (𝜑 → 𝐴 ∈ ( − “ (𝑃 × 𝑃)))
15		fvelima 6248	. . 3 ⊢ ((Fun − ∧ 𝐴 ∈ ( − “ (𝑃 × 𝑃))) → ∃𝑎 ∈ (𝑃 × 𝑃)( − ‘𝑎) = 𝐴)
16	11, 14, 15	syl2anc 693	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑃 × 𝑃)( − ‘𝑎) = 𝐴)
17	10, 16	r19.29a 3078	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝐴 = (𝑥 − 𝑦))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  ⟨cop 4183   × cxp 5112   “ cima 5117  Fun wfun 5882  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  ≤Gcleg 25477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  legso  25494