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| Mirrors > Home > MPE Home > Th. List > dvgt0lem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for dvgt0 26043 and dvlt0 26044. (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 3202 | . . . 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 13469 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
| 7 | 4, 5, 6 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 8 | ltso 11341 | . . . . . 6 ⊢ < Or ℝ | |
| 9 | soss 5612 | . . . . . 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 24919 | . . . . . 6 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ) | |
| 15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
| 16 | ssidd 4007 | . . . . 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 6736 | . . . . 5 ⊢ (𝐹:(𝐴[,]𝐵)⟶ℝ → 𝐹 Fn (𝐴[,]𝐵)) | |
| 21 | 13, 14, 20 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵)) |
| 22 | fnresdm 6687 | . . . 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 6698 | . . 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 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 class class class wbr 5143 Or wor 5591 ran crn 5686 ↾ cres 5687 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 Isom wiso 6562 (class class class)co 7431 ℝcr 11154 < clt 11295 (,)cioo 13387 [,]cicc 13390 –cn→ccncf 24902 D cdv 25898 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-icc 13394 df-cncf 24904 |
| This theorem is referenced by: dvgt0 26043 dvlt0 26044 |
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