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Theorem legtrd 28274

Description: Transitivity of the less-than relationship. Proposition 5.8 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-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)
legid.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
legid.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
legtrd.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
legtrd.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
legtrd.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
legtrd.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
legtrd.1	⊢ (𝜑 → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
legtrd.2	⊢ (𝜑 → (𝐶 − 𝐷) ≤ (𝐸 − 𝐹))

**Assertion**
Ref	Expression
legtrd	⊢ (𝜑 → (𝐴 − 𝐵) ≤ (𝐸 − 𝐹))

Proof of Theorem legtrd Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		legval.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2731	. . . . . 6 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
3		legval.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
4		legval.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad4antr 729	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → 𝐺 ∈ TarskiG)
6		legtrd.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7	6	ad4antr 729	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → 𝐶 ∈ 𝑃)
8		legtrd.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
9	8	ad4antr 729	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → 𝐷 ∈ 𝑃)
10		simp-4r 781	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → 𝑥 ∈ 𝑃)
11		eqid 2731	. . . . . 6 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
12		legtrd.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
13	12	ad4antr 729	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → 𝐸 ∈ 𝑃)
14		simplr 766	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → 𝑦 ∈ 𝑃)
15		legval.d	. . . . . 6 ⊢ − = (dist‘𝐺)
16		simpllr 773	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥)))
17	16	simpld 494	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → 𝑥 ∈ (𝐶𝐼𝐷))
18	1, 2, 3, 5, 7, 10, 9, 17	btwncolg3 28242	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → (𝐷 ∈ (𝐶(LineG‘𝐺)𝑥) ∨ 𝐶 = 𝑥))
19		simprr 770	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → (𝐶 − 𝐷) = (𝐸 − 𝑦))
20	1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15, 18, 19	lnext 28252	. . . . 5 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → ∃𝑧 ∈ 𝑃 ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩)
21	5	ad2antrr 723	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝐺 ∈ TarskiG)
22	13	ad2antrr 723	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝐸 ∈ 𝑃)
23		simplr 766	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝑧 ∈ 𝑃)
24		simp-4r 781	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝑦 ∈ 𝑃)
25		legtrd.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
26	25	ad6antr 733	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝐹 ∈ 𝑃)
27	7	ad2antrr 723	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝐶 ∈ 𝑃)
28	10	ad2antrr 723	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝑥 ∈ 𝑃)
29	9	ad2antrr 723	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝐷 ∈ 𝑃)
30		simpr 484	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩)
31	1, 15, 3, 11, 21, 27, 29, 28, 22, 24, 23, 30	cgr3swap23 28209	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → ⟨“𝐶𝑥𝐷”⟩(cgrG‘𝐺)⟨“𝐸𝑧𝑦”⟩)
32	17	ad2antrr 723	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝑥 ∈ (𝐶𝐼𝐷))
33	1, 15, 3, 11, 21, 27, 28, 29, 22, 23, 24, 31, 32	tgbtwnxfr 28215	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝑧 ∈ (𝐸𝐼𝑦))
34		simpllr 773	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦)))
35	34	simpld 494	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝑦 ∈ (𝐸𝐼𝐹))
36	1, 15, 3, 21, 22, 23, 24, 26, 33, 35	tgbtwnexch 28183	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → 𝑧 ∈ (𝐸𝐼𝐹))
37		simp-5r 783	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥)))
38	37	simprd 495	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → (𝐴 − 𝐵) = (𝐶 − 𝑥))
39	1, 15, 3, 11, 21, 27, 28, 29, 22, 23, 24, 31	cgr3simp1 28205	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → (𝐶 − 𝑥) = (𝐸 − 𝑧))
40	38, 39	eqtrd 2771	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → (𝐴 − 𝐵) = (𝐸 − 𝑧))
41	36, 40	jca 511	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) ∧ ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩) → (𝑧 ∈ (𝐸𝐼𝐹) ∧ (𝐴 − 𝐵) = (𝐸 − 𝑧)))
42	41	ex 412	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) ∧ 𝑧 ∈ 𝑃) → (⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩ → (𝑧 ∈ (𝐸𝐼𝐹) ∧ (𝐴 − 𝐵) = (𝐸 − 𝑧))))
43	42	reximdva 3167	. . . . 5 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → (∃𝑧 ∈ 𝑃 ⟨“𝐶𝐷𝑥”⟩(cgrG‘𝐺)⟨“𝐸𝑦𝑧”⟩ → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐸𝐼𝐹) ∧ (𝐴 − 𝐵) = (𝐸 − 𝑧))))
44	20, 43	mpd 15	. . . 4 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) ∧ 𝑦 ∈ 𝑃) ∧ (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐸𝐼𝐹) ∧ (𝐴 − 𝐵) = (𝐸 − 𝑧)))
45		legtrd.2	. . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐷) ≤ (𝐸 − 𝐹))
46		legval.l	. . . . . . 7 ⊢ ≤ = (≤G‘𝐺)
47	1, 15, 3, 46, 4, 6, 8, 12, 25	legov 28270	. . . . . 6 ⊢ (𝜑 → ((𝐶 − 𝐷) ≤ (𝐸 − 𝐹) ↔ ∃𝑦 ∈ 𝑃 (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦))))
48	45, 47	mpbid 231	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦)))
49	48	ad2antrr 723	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) → ∃𝑦 ∈ 𝑃 (𝑦 ∈ (𝐸𝐼𝐹) ∧ (𝐶 − 𝐷) = (𝐸 − 𝑦)))
50	44, 49	r19.29a 3161	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐸𝐼𝐹) ∧ (𝐴 − 𝐵) = (𝐸 − 𝑧)))
51		legtrd.1	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
52		legid.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
53		legid.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
54	1, 15, 3, 46, 4, 52, 53, 6, 8	legov 28270	. . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ↔ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥))))
55	51, 54	mpbid 231	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑥)))
56	50, 55	r19.29a 3161	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐸𝐼𝐹) ∧ (𝐴 − 𝐵) = (𝐸 − 𝑧)))
57	1, 15, 3, 46, 4, 52, 53, 12, 25	legov 28270	. 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐸 − 𝐹) ↔ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐸𝐼𝐹) ∧ (𝐴 − 𝐵) = (𝐸 − 𝑧))))
58	56, 57	mpbird 257	1 ⊢ (𝜑 → (𝐴 − 𝐵) ≤ (𝐸 − 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ⟨“cs3 14800 Basecbs 17151 distcds 17213 TarskiGcstrkg 28112 Itvcitv 28118 LineGclng 28119 cgrGccgrg 28195 ≤Gcleg 28267

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-er 8709 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-xnn0 12552 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-hash 14298 df-word 14472 df-concat 14528 df-s1 14553 df-s2 14806 df-s3 14807 df-trkgc 28133 df-trkgb 28134 df-trkgcb 28135 df-trkg 28138 df-cgrg 28196 df-leg 28268

This theorem is referenced by: legso 28284

W3C validator