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Theorem legov 28101

Description: Value of the less-than relationship. Definition 5.4 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 21-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)
legov.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
legov.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
legov.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
legov.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)

**Assertion**
Ref	Expression
legov	⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ↔ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))))

Distinct variable groups: 𝑧, − 𝑧,𝐴 𝑧,𝐵 𝑧,𝐶 𝑧,𝐷 𝑧,𝐼 𝑧,𝑃 𝑧,𝐺 𝜑,𝑧 Allowed substitution hint: ≤ (𝑧)

Proof of Theorem legov Dummy variables 𝑐 𝑑 𝑒 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		legval.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		legval.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		legval.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		legval.l	. . . . 5 ⊢ ≤ = (≤G‘𝐺)
5		legval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	1, 2, 3, 4, 5	legval 28100	. . . 4 ⊢ (𝜑 → ≤ = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))})
7	6	breqd 5160	. . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ↔ (𝐴 − 𝐵){⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))} (𝐶 − 𝐷)))
8		ovex 7446	. . . 4 ⊢ (𝐴 − 𝐵) ∈ V
9		ovex 7446	. . . 4 ⊢ (𝐶 − 𝐷) ∈ V
10		simpr 483	. . . . . . 7 ⊢ ((𝑒 = (𝐴 − 𝐵) ∧ 𝑓 = (𝐶 − 𝐷)) → 𝑓 = (𝐶 − 𝐷))
11	10	eqeq1d 2732	. . . . . 6 ⊢ ((𝑒 = (𝐴 − 𝐵) ∧ 𝑓 = (𝐶 − 𝐷)) → (𝑓 = (𝑥 − 𝑦) ↔ (𝐶 − 𝐷) = (𝑥 − 𝑦)))
12		simpl 481	. . . . . . . . 9 ⊢ ((𝑒 = (𝐴 − 𝐵) ∧ 𝑓 = (𝐶 − 𝐷)) → 𝑒 = (𝐴 − 𝐵))
13	12	eqeq1d 2732	. . . . . . . 8 ⊢ ((𝑒 = (𝐴 − 𝐵) ∧ 𝑓 = (𝐶 − 𝐷)) → (𝑒 = (𝑥 − 𝑧) ↔ (𝐴 − 𝐵) = (𝑥 − 𝑧)))
14	13	anbi2d 627	. . . . . . 7 ⊢ ((𝑒 = (𝐴 − 𝐵) ∧ 𝑓 = (𝐶 − 𝐷)) → ((𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)) ↔ (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
15	14	rexbidv 3176	. . . . . 6 ⊢ ((𝑒 = (𝐴 − 𝐵) ∧ 𝑓 = (𝐶 − 𝐷)) → (∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)) ↔ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
16	11, 15	anbi12d 629	. . . . 5 ⊢ ((𝑒 = (𝐴 − 𝐵) ∧ 𝑓 = (𝐶 − 𝐷)) → ((𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧))) ↔ ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)))))
17	16	2rexbidv 3217	. . . 4 ⊢ ((𝑒 = (𝐴 − 𝐵) ∧ 𝑓 = (𝐶 − 𝐷)) → (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧))) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)))))
18		eqid 2730	. . . 4 ⊢ {⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))} = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))}
19	8, 9, 17, 18	braba 5538	. . 3 ⊢ ((𝐴 − 𝐵){⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))} (𝐶 − 𝐷) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
20	7, 19	bitrdi 286	. 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)))))
21		anass 467	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧))) ↔ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ ((𝐶 − 𝐷) = (𝑐 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)))))
22	21	anbi1i 622	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧))) ∧ 𝑥 ∈ 𝑃) ↔ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ ((𝐶 − 𝐷) = (𝑐 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)))) ∧ 𝑥 ∈ 𝑃))
23		eqid 2730	. . . . . . . . . . 11 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
24	5	ad5antr 730	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → 𝐺 ∈ TarskiG)
25	24	adantr 479	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → 𝐺 ∈ TarskiG)
26		simp-5r 782	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → 𝑐 ∈ 𝑃)
27	26	adantr 479	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → 𝑐 ∈ 𝑃)
28		simpllr 772	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → 𝑥 ∈ 𝑃)
29		simp-4r 780	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → 𝑑 ∈ 𝑃)
30	29	adantr 479	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → 𝑑 ∈ 𝑃)
31		legov.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
32	31	ad5antr 730	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → 𝐶 ∈ 𝑃)
33	32	adantr 479	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → 𝐶 ∈ 𝑃)
34		simprl 767	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → 𝑧 ∈ 𝑃)
35		legov.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
36	35	ad5antr 730	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → 𝐷 ∈ 𝑃)
37	36	adantr 479	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → 𝐷 ∈ 𝑃)
38		simprr 769	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)
39	1, 2, 3, 23, 25, 27, 30, 28, 33, 37, 34, 38	cgr3swap23 28040	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → ⟨“𝑐𝑥𝑑”⟩(cgrG‘𝐺)⟨“𝐶𝑧𝐷”⟩)
40		simprl 767	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → 𝑥 ∈ (𝑐𝐼𝑑))
41	40	adantr 479	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → 𝑥 ∈ (𝑐𝐼𝑑))
42	1, 2, 3, 23, 25, 27, 28, 30, 33, 34, 37, 39, 41	tgbtwnxfr 28046	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → 𝑧 ∈ (𝐶𝐼𝐷))
43		simplrr 774	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → (𝐴 − 𝐵) = (𝑐 − 𝑥))
44	1, 2, 3, 23, 25, 27, 28, 30, 33, 34, 37, 39	cgr3simp1 28036	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → (𝑐 − 𝑥) = (𝐶 − 𝑧))
45	43, 44	eqtrd 2770	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → (𝐴 − 𝐵) = (𝐶 − 𝑧))
46	42, 45	jca 510	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) ∧ (𝑧 ∈ 𝑃 ∧ ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)) → (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))
47		eqid 2730	. . . . . . . . . 10 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
48		simplr 765	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → 𝑥 ∈ 𝑃)
49	1, 47, 3, 24, 26, 48, 29, 40	btwncolg3 28073	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → (𝑑 ∈ (𝑐(LineG‘𝐺)𝑥) ∨ 𝑐 = 𝑥))
50		simpllr 772	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → (𝐶 − 𝐷) = (𝑐 − 𝑑))
51	50	eqcomd 2736	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → (𝑐 − 𝑑) = (𝐶 − 𝐷))
52	1, 47, 3, 24, 26, 29, 48, 23, 32, 36, 2, 49, 51	lnext 28083	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → ∃𝑧 ∈ 𝑃 ⟨“𝑐𝑑𝑥”⟩(cgrG‘𝐺)⟨“𝐶𝐷𝑧”⟩)
53	46, 52	reximddv 3169	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))
54	53	adantllr 715	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ (𝐶 − 𝐷) = (𝑐 − 𝑑)) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))
55	22, 54	sylanbr 580	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ ((𝐶 − 𝐷) = (𝑐 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)))) ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))
56		simprr 769	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ ((𝐶 − 𝐷) = (𝑐 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)))
57		eleq1w 2814	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑐𝐼𝑑) ↔ 𝑧 ∈ (𝑐𝐼𝑑)))
58		oveq2 7421	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑐 − 𝑥) = (𝑐 − 𝑧))
59	58	eqeq2d 2741	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝐴 − 𝐵) = (𝑐 − 𝑥) ↔ (𝐴 − 𝐵) = (𝑐 − 𝑧)))
60	57, 59	anbi12d 629	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥)) ↔ (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧))))
61	60	cbvrexvw 3233	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥)) ↔ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)))
62	56, 61	sylibr 233	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ ((𝐶 − 𝐷) = (𝑐 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)))) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑥)))
63	55, 62	r19.29a 3160	. . . . 5 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ ((𝐶 − 𝐷) = (𝑐 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))
64	63	adantl3r 746	. . . 4 ⊢ (((((𝜑 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)))) ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 ∈ 𝑃) ∧ ((𝐶 − 𝐷) = (𝑐 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))
65		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)))) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
66		oveq1 7420	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (𝑐 − 𝑑) = (𝑥 − 𝑑))
67	66	eqeq2d 2741	. . . . . . 7 ⊢ (𝑐 = 𝑥 → ((𝐶 − 𝐷) = (𝑐 − 𝑑) ↔ (𝐶 − 𝐷) = (𝑥 − 𝑑)))
68		oveq1 7420	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → (𝑐𝐼𝑑) = (𝑥𝐼𝑑))
69	68	eleq2d 2817	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (𝑧 ∈ (𝑐𝐼𝑑) ↔ 𝑧 ∈ (𝑥𝐼𝑑)))
70		oveq1 7420	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → (𝑐 − 𝑧) = (𝑥 − 𝑧))
71	70	eqeq2d 2741	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → ((𝐴 − 𝐵) = (𝑐 − 𝑧) ↔ (𝐴 − 𝐵) = (𝑥 − 𝑧)))
72	69, 71	anbi12d 629	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → ((𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)) ↔ (𝑧 ∈ (𝑥𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
73	72	rexbidv 3176	. . . . . . 7 ⊢ (𝑐 = 𝑥 → (∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧)) ↔ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
74	67, 73	anbi12d 629	. . . . . 6 ⊢ (𝑐 = 𝑥 → (((𝐶 − 𝐷) = (𝑐 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧))) ↔ ((𝐶 − 𝐷) = (𝑥 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)))))
75		oveq2 7421	. . . . . . . 8 ⊢ (𝑑 = 𝑦 → (𝑥 − 𝑑) = (𝑥 − 𝑦))
76	75	eqeq2d 2741	. . . . . . 7 ⊢ (𝑑 = 𝑦 → ((𝐶 − 𝐷) = (𝑥 − 𝑑) ↔ (𝐶 − 𝐷) = (𝑥 − 𝑦)))
77		oveq2 7421	. . . . . . . . . 10 ⊢ (𝑑 = 𝑦 → (𝑥𝐼𝑑) = (𝑥𝐼𝑦))
78	77	eleq2d 2817	. . . . . . . . 9 ⊢ (𝑑 = 𝑦 → (𝑧 ∈ (𝑥𝐼𝑑) ↔ 𝑧 ∈ (𝑥𝐼𝑦)))
79	78	anbi1d 628	. . . . . . . 8 ⊢ (𝑑 = 𝑦 → ((𝑧 ∈ (𝑥𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)) ↔ (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
80	79	rexbidv 3176	. . . . . . 7 ⊢ (𝑑 = 𝑦 → (∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)) ↔ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
81	76, 80	anbi12d 629	. . . . . 6 ⊢ (𝑑 = 𝑦 → (((𝐶 − 𝐷) = (𝑥 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))) ↔ ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)))))
82	74, 81	cbvrex2vw 3237	. . . . 5 ⊢ (∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑐 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧))) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
83	65, 82	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)))) → ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑐 − 𝑑) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑐𝐼𝑑) ∧ (𝐴 − 𝐵) = (𝑐 − 𝑧))))
84	64, 83	r19.29vva 3211	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))
85	31	adantr 479	. . . 4 ⊢ ((𝜑 ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))) → 𝐶 ∈ 𝑃)
86	35	adantr 479	. . . 4 ⊢ ((𝜑 ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))) → 𝐷 ∈ 𝑃)
87		eqidd 2731	. . . 4 ⊢ ((𝜑 ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))) → (𝐶 − 𝐷) = (𝐶 − 𝐷))
88		simpr 483	. . . 4 ⊢ ((𝜑 ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))
89		oveq1 7420	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 − 𝑦) = (𝐶 − 𝑦))
90	89	eqeq2d 2741	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐶 − 𝐷) = (𝑥 − 𝑦) ↔ (𝐶 − 𝐷) = (𝐶 − 𝑦)))
91		oveq1 7420	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑥𝐼𝑦) = (𝐶𝐼𝑦))
92	91	eleq2d 2817	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝐶𝐼𝑦)))
93		oveq1 7420	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑥 − 𝑧) = (𝐶 − 𝑧))
94	93	eqeq2d 2741	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → ((𝐴 − 𝐵) = (𝑥 − 𝑧) ↔ (𝐴 − 𝐵) = (𝐶 − 𝑧)))
95	92, 94	anbi12d 629	. . . . . . 7 ⊢ (𝑥 = 𝐶 → ((𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)) ↔ (𝑧 ∈ (𝐶𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))))
96	95	rexbidv 3176	. . . . . 6 ⊢ (𝑥 = 𝐶 → (∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧)) ↔ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))))
97	90, 96	anbi12d 629	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))) ↔ ((𝐶 − 𝐷) = (𝐶 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))))
98		oveq2 7421	. . . . . . 7 ⊢ (𝑦 = 𝐷 → (𝐶 − 𝑦) = (𝐶 − 𝐷))
99	98	eqeq2d 2741	. . . . . 6 ⊢ (𝑦 = 𝐷 → ((𝐶 − 𝐷) = (𝐶 − 𝑦) ↔ (𝐶 − 𝐷) = (𝐶 − 𝐷)))
100		oveq2 7421	. . . . . . . . 9 ⊢ (𝑦 = 𝐷 → (𝐶𝐼𝑦) = (𝐶𝐼𝐷))
101	100	eleq2d 2817	. . . . . . . 8 ⊢ (𝑦 = 𝐷 → (𝑧 ∈ (𝐶𝐼𝑦) ↔ 𝑧 ∈ (𝐶𝐼𝐷)))
102	101	anbi1d 628	. . . . . . 7 ⊢ (𝑦 = 𝐷 → ((𝑧 ∈ (𝐶𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)) ↔ (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))))
103	102	rexbidv 3176	. . . . . 6 ⊢ (𝑦 = 𝐷 → (∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)) ↔ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))))
104	99, 103	anbi12d 629	. . . . 5 ⊢ (𝑦 = 𝐷 → (((𝐶 − 𝐷) = (𝐶 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))) ↔ ((𝐶 − 𝐷) = (𝐶 − 𝐷) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))))
105	97, 104	rspc2ev 3625	. . . 4 ⊢ ((𝐶 ∈ 𝑃 ∧ 𝐷 ∈ 𝑃 ∧ ((𝐶 − 𝐷) = (𝐶 − 𝐷) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧)))) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
106	85, 86, 87, 88, 105	syl112anc 1372	. . 3 ⊢ ((𝜑 ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))))
107	84, 106	impbida 797	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝐶 − 𝐷) = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝐵) = (𝑥 − 𝑧))) ↔ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))))
108	20, 107	bitrd 278	1 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ↔ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐵) = (𝐶 − 𝑧))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∃wrex 3068 class class class wbr 5149 {copab 5211 ‘cfv 6544 (class class class)co 7413 ⟨“cs3 14799 Basecbs 17150 distcds 17212 TarskiGcstrkg 27943 Itvcitv 27949 LineGclng 27950 cgrGccgrg 28026 ≤Gcleg 28098

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oadd 8474 df-er 8707 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-dju 9900 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-2 12281 df-3 12282 df-n0 12479 df-xnn0 12551 df-z 12565 df-uz 12829 df-fz 13491 df-fzo 13634 df-hash 14297 df-word 14471 df-concat 14527 df-s1 14552 df-s2 14805 df-s3 14806 df-trkgc 27964 df-trkgb 27965 df-trkgcb 27966 df-trkg 27969 df-cgrg 28027 df-leg 28099

This theorem is referenced by: legov2 28102 legid 28103 btwnleg 28104 legtrd 28105 legtri3 28106 legtrid 28107 leg0 28108 mideulem 28252 opphllem3 28265

W3C validator