![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dvgt0lem2 | Structured version Visualization version GIF version |
Description: Lemma for dvgt0 25981 and dvlt0 25982. (Contributed by Mario Carneiro, 19-Feb-2015.) |
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)) |
Ref | Expression |
---|---|
dvgt0lem2 | ⊢ (𝜑 → 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹)) |
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 411 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (𝐹‘𝑥)𝑂(𝐹‘𝑦))) |
3 | 2 | ralrimivva 3190 | . . . 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 13441 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
7 | 4, 5, 6 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
8 | ltso 11326 | . . . . . 6 ⊢ < Or ℝ | |
9 | soss 5610 | . . . . . 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 24857 | . . . . . 6 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
16 | ssidd 4000 | . . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵)) | |
17 | soisores 7334 | . . . . 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 837 | . . . 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 256 | . . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵)))) |
20 | ffn 6723 | . . . . 5 ⊢ (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵)) | |
21 | 13, 14, 20 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵)) |
22 | fnresdm 6675 | . . . 4 ⊢ (𝐹 Fn (𝐴[,]𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) = 𝐹) | |
23 | isoeq1 7324 | . . . 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 231 | . 2 ⊢ (𝜑 → 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), (𝐹 “ (𝐴[,]𝐵)))) |
26 | fnima 6686 | . . 3 ⊢ (𝐹 Fn (𝐴[,]𝐵) → (𝐹 “ (𝐴[,]𝐵)) = ran 𝐹) | |
27 | isoeq5 7328 | . . 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 231 | 1 ⊢ (𝜑 → 𝐹 Isom < , 𝑂 ((𝐴[,]𝐵), ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ⊆ wss 3944 class class class wbr 5149 Or wor 5589 ran crn 5679 ↾ cres 5680 “ cima 5681 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 Isom wiso 6550 (class class class)co 7419 ℝcr 11139 < clt 11280 (,)cioo 13359 [,]cicc 13362 –cn→ccncf 24840 D cdv 25836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-icc 13366 df-cncf 24842 |
This theorem is referenced by: dvgt0 25981 dvlt0 25982 |
Copyright terms: Public domain | W3C validator |