Theorem legval 28890

Description: Value of the less-than relationship. (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)

Assertion

Ref	Expression
legval	⊢ (𝜑 → ≤ = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))})

Distinct variable groups:   𝑒,𝑓,𝐺   𝑥,𝑦,𝑧,𝐼   𝑥,𝑒,𝑦,𝑧,𝑃,𝑓   − ,𝑒,𝑓,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑒, 𝑓)   𝐺(𝑥, 𝑦, 𝑧)   𝐼(𝑒, 𝑓)   ≤ (𝑥, 𝑦, 𝑧, 𝑒, 𝑓)

Proof of Theorem legval

Dummy variables 𝑑 𝑔 𝑖 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		legval.l	. 2 ⊢ ≤ = (≤G‘𝐺)
2		legval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
3		elex 3479	. . 3 ⊢ (𝐺 ∈ TarskiG → 𝐺 ∈ V)
4		legval.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
5		legval.d	. . . . . 6 ⊢ − = (dist‘𝐺)
6		legval.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
7		simp1 1154	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → 𝑝 = 𝑃)
8	7	eqcomd 2772	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → 𝑃 = 𝑝)
9		simp2 1155	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → 𝑑 = − )
10	9	eqcomd 2772	. . . . . . . . . . 11 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → − = 𝑑)
11	10	oveqd 7440	. . . . . . . . . 10 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → (𝑥 − 𝑦) = (𝑥𝑑𝑦))
12	11	eqeq2d 2777	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → (𝑓 = (𝑥 − 𝑦) ↔ 𝑓 = (𝑥𝑑𝑦)))
13		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
14	13	eqcomd 2772	. . . . . . . . . . . . 13 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → 𝐼 = 𝑖)
15	14	oveqd 7440	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → (𝑥𝐼𝑦) = (𝑥𝑖𝑦))
16	15	eleq2d 2852	. . . . . . . . . . 11 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥𝑖𝑦)))
17	10	oveqd 7440	. . . . . . . . . . . 12 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → (𝑥 − 𝑧) = (𝑥𝑑𝑧))
18	17	eqeq2d 2777	. . . . . . . . . . 11 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → (𝑒 = (𝑥 − 𝑧) ↔ 𝑒 = (𝑥𝑑𝑧)))
19	16, 18	anbi12d 644	. . . . . . . . . 10 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → ((𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)) ↔ (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧))))
20	8, 19	rexeqbidv 3342	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → (∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)) ↔ ∃𝑧 ∈ 𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧))))
21	12, 20	anbi12d 644	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → ((𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧))) ↔ (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧 ∈ 𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))))
22	8, 21	rexeqbidv 3342	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → (∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧))) ↔ ∃𝑦 ∈ 𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧 ∈ 𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))))
23	8, 22	rexeqbidv 3342	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ∧ 𝑖 = 𝐼) → (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧))) ↔ ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧 ∈ 𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))))
24	4, 5, 6, 23	sbcie3s 17247	. . . . 5 ⊢ (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧 ∈ 𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧))) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))))
25	24	opabbidv 5182	. . . 4 ⊢ (𝑔 = 𝐺 → {⟨𝑒, 𝑓⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧 ∈ 𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))} = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))})
26		df-leg 28889	. . . 4 ⊢ ≤G = (𝑔 ∈ V ↦ {⟨𝑒, 𝑓⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧 ∈ 𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))})
27	5	fvexi 6902	. . . . . . . . 9 ⊢ − ∈ V
28	27	imaex 7920	. . . . . . . 8 ⊢ ( − “ (𝑃 × 𝑃)) ∈ V
29		p0ex 5360	. . . . . . . 8 ⊢ {∅} ∈ V
30	28, 29	unex 7755	. . . . . . 7 ⊢ (( − “ (𝑃 × 𝑃)) ∪ {∅}) ∈ V
31	30	a1i 11	. . . . . 6 ⊢ (⊤ → (( − “ (𝑃 × 𝑃)) ∪ {∅}) ∈ V)
32		simprr 785	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑))) → 𝑒 = (𝑥 − 𝑑))
33		ovima0 7602	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑃 ∧ 𝑑 ∈ 𝑃) → (𝑥 − 𝑑) ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}))
34	33	ad5ant14 770	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑))) → (𝑥 − 𝑑) ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}))
35	32, 34	eqeltrd 2866	. . . . . . . . . . . 12 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑))) → 𝑒 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}))
36		simpllr 788	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑))) → (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧))))
37	36	simpld 500	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑))) → 𝑓 = (𝑥 − 𝑦))
38		ovima0 7602	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) → (𝑥 − 𝑦) ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}))
39	38	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑))) → (𝑥 − 𝑦) ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}))
40	37, 39	eqeltrd 2866	. . . . . . . . . . . 12 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑))) → 𝑓 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}))
41	35, 40	jca 521	. . . . . . . . . . 11 ⊢ (((((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑))) → (𝑒 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅})))
42		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) → ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))
43		eleq1w 2849	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑑 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑑 ∈ (𝑥𝐼𝑦)))
44		oveq2 7431	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑑 → (𝑥 − 𝑧) = (𝑥 − 𝑑))
45	44	eqeq2d 2777	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑑 → (𝑒 = (𝑥 − 𝑧) ↔ 𝑒 = (𝑥 − 𝑑)))
46	43, 45	anbi12d 644	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑑 → ((𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)) ↔ (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑))))
47	46	cbvrexvw 3247	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)) ↔ ∃𝑑 ∈ 𝑃 (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑)))
48	42, 47	sylib 221	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) → ∃𝑑 ∈ 𝑃 (𝑑 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑑)))
49	41, 48	r19.29a 3176	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) ∧ (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) → (𝑒 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅})))
50	49	ex 418	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃) → ((𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧))) → (𝑒 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}))))
51	50	rexlimivv 3210	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧))) → (𝑒 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅})))
52	51	adantl 487	. . . . . . 7 ⊢ ((⊤ ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) → (𝑒 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}) ∧ 𝑓 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅})))
53	52	simpld 500	. . . . . 6 ⊢ ((⊤ ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) → 𝑒 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}))
54	52	simprd 501	. . . . . 6 ⊢ ((⊤ ∧ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))) → 𝑓 ∈ (( − “ (𝑃 × 𝑃)) ∪ {∅}))
55	31, 31, 53, 54	opabex2 8063	. . . . 5 ⊢ (⊤ → {⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))} ∈ V)
56	55	mptru 1577	. . . 4 ⊢ {⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))} ∈ V
57	25, 26, 56	fvmpt 6996	. . 3 ⊢ (𝐺 ∈ V → (≤G‘𝐺) = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))})
58	2, 3, 57	3syl 19	. 2 ⊢ (𝜑 → (≤G‘𝐺) = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))})
59	1, 58	eqtrid 2813	1 ⊢ (𝜑 → ≤ = {⟨𝑒, 𝑓⟩ ∣ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (𝑓 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ 𝑃 (𝑧 ∈ (𝑥𝐼𝑦) ∧ 𝑒 = (𝑥 − 𝑧)))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  ∃wrex 3092  Vcvv 3458  [wsbc 3747   ∪ cun 3906  ∅c0 4289  {csn 4594  {copab 5178   × cxp 5664   “ cima 5669  ‘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-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  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  df-leg 28889

This theorem is used by:  legov  28891