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Mirrors > Home > MPE Home > Th. List > dvgt0lem2 | Structured version Visualization version GIF version |
Description: Lemma for dvgt0 26058 and dvlt0 26059. (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 412 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) → (𝑥 < 𝑦 → (𝐹‘𝑥)𝑂(𝐹‘𝑦))) |
3 | 2 | ralrimivva 3200 | . . . 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 13466 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
7 | 4, 5, 6 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
8 | ltso 11339 | . . . . . 6 ⊢ < Or ℝ | |
9 | soss 5617 | . . . . . 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 24933 | . . . . . 6 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
16 | ssidd 4019 | . . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐵)) | |
17 | soisores 7347 | . . . . 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 839 | . . . 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 6737 | . . . . 5 ⊢ (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵)) | |
21 | 13, 14, 20 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵)) |
22 | fnresdm 6688 | . . . 4 ⊢ (𝐹 Fn (𝐴[,]𝐵) → (𝐹 ↾ (𝐴[,]𝐵)) = 𝐹) | |
23 | isoeq1 7337 | . . . 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 6699 | . . 3 ⊢ (𝐹 Fn (𝐴[,]𝐵) → (𝐹 “ (𝐴[,]𝐵)) = ran 𝐹) | |
27 | isoeq5 7341 | . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 class class class wbr 5148 Or wor 5596 ran crn 5690 ↾ cres 5691 “ cima 5692 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 Isom wiso 6564 (class class class)co 7431 ℝcr 11152 < clt 11293 (,)cioo 13384 [,]cicc 13387 –cn→ccncf 24916 D cdv 25913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-icc 13391 df-cncf 24918 |
This theorem is referenced by: dvgt0 26058 dvlt0 26059 |
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