Theorem dvgt0lem2 25915

Description: Lemma for dvgt0 25916 and dvlt0 25917. (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 3181	. . . 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 13397	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
7	4, 5, 6	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
8		ltso 11261	. . . . . 6 ⊢ < Or ℝ
9		soss 5569	. . . . . 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 24793	. . . . . 6 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
16		ssidd 3973	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵))
17		soisores 7305	. . . . 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 6691	. . . . 5 ⊢ (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵))
21	13, 14, 20	3syl 18	. . . 4 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
22		fnresdm 6640	. . . 4 ⊢ (𝐹 Fn (𝐴[,]𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) = 𝐹)
23		isoeq1 7295	. . . 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 6651	. . 3 ⊢ (𝐹 Fn (𝐴[,]𝐵) → (𝐹 “ (𝐴[,]𝐵)) = ran 𝐹)
27		isoeq5 7299	. . 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 1540   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3917   class class class wbr 5110   Or wor 5548  ran crn 5642   ↾ cres 5643   “ cima 5644   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514   Isom wiso 6515  (class class class)co 7390  ℝcr 11074   < clt 11215  (,)cioo 13313  [,]cicc 13316  –cn→ccncf 24776   D cdv 25771

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-pre-lttri 11149  ax-pre-lttrn 11150

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-icc 13320  df-cncf 24778

This theorem is referenced by:  dvgt0  25916  dvlt0  25917