Theorem dvgt0lem2 25935

Description: Lemma for dvgt0 25936 and dvlt0 25937. (Contributed by Mario Carneiro, 19-Feb-2015.)

Hypotheses

Ref	Expression
dvgt0.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvgt0.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dvgt0.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
dvgt0lem.d	⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶𝑆)
dvgt0lem.o	⊢ 𝑂 Or ℝ
dvgt0lem.i	⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥)𝑂(𝐹‘𝑦))

Assertion

Ref	Expression
dvgt0lem2	⊢ (𝜑 → 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑂,𝑦   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Allowed substitution hints:   𝑆(𝑥,𝑦)

Proof of Theorem dvgt0lem2

Step	Hyp	Ref	Expression
1		dvgt0lem.i	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥)𝑂(𝐹‘𝑦))
2	1	ex 412	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (𝐹‘𝑥)𝑂(𝐹‘𝑦)))
3	2	ralrimivva 3175	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥)𝑂(𝐹‘𝑦)))
4		dvgt0.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
5		dvgt0.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
6		iccssre 13329	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
7	4, 5, 6	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8		ltso 11193	. . . . . 6 ⊢ < Or ℝ
9		soss 5542	. . . . . 6 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → ( < Or ℝ → < Or (𝐴[,]𝐵)))
10	7, 8, 9	mpisyl 21	. . . . 5 ⊢ (𝜑 → < Or (𝐴[,]𝐵))
11		dvgt0lem.o	. . . . . 6 ⊢ 𝑂 Or ℝ
12	11	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑂 Or ℝ)
13		dvgt0.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
14		cncff 24813	. . . . . 6 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
16		ssidd 3953	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵))
17		soisores 7261	. . . . 5 ⊢ ((( < Or (𝐴[,]𝐵) ∧ 𝑂 Or ℝ) ∧ (𝐹:(𝐴[,]𝐵)⟶ℝ ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵))) → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥)𝑂(𝐹‘𝑦))))
18	10, 12, 15, 16, 17	syl22anc 838	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥)𝑂(𝐹‘𝑦))))
19	3, 18	mpbird 257	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
20		ffn 6651	. . . . 5 ⊢ (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵))
21	13, 14, 20	3syl 18	. . . 4 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
22		fnresdm 6600	. . . 4 ⊢ (𝐹 Fn (𝐴[,]𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) = 𝐹)
23		isoeq1 7251	. . . 4 ⊢ ((𝐹 ↾ (𝐴[,]𝐵)) = 𝐹 → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵)))))
24	21, 22, 23	3syl 18	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵)))))
25	19, 24	mpbid 232	. 2 ⊢ (𝜑 → 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))))
26		fnima 6611	. . 3 ⊢ (𝐹 Fn (𝐴[,]𝐵) → (𝐹 “ (𝐴[,]𝐵)) = ran 𝐹)
27		isoeq5 7255	. . 3 ⊢ ((𝐹 “ (𝐴[,]𝐵)) = ran 𝐹 → (𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹)))
28	21, 26, 27	3syl 18	. 2 ⊢ (𝜑 → (𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵))) ↔ 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹)))
29	25, 28	mpbid 232	1 ⊢ (𝜑 → 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897   class class class wbr 5089   Or wor 5521  ran crn 5615   ↾ cres 5616   “ cima 5617   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481   Isom wiso 6482  (class class class)co 7346  ℝcr 11005   < clt 11146  (,)cioo 13245  [,]cicc 13248  –cn→ccncf 24796   D cdv 25791

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-pre-lttri 11080  ax-pre-lttrn 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-po 5522  df-so 5523  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-ov 7349  df-oprab 7350  df-mpo 7351  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-icc 13252  df-cncf 24798

This theorem is referenced by:  dvgt0  25936  dvlt0  25937