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Mirrors > Home > MPE Home > Th. List > ivthlem1 | Structured version Visualization version GIF version |
Description: Lemma for ivth 23030. The set 𝑆 of all 𝑥 values with (𝐹‘𝑥) less than 𝑈 is lower bounded by 𝐴 and upper bounded by 𝐵. (Contributed by Mario Carneiro, 17-Jun-2014.) |
Ref | Expression |
---|---|
ivth.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ivth.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ivth.3 | ⊢ (𝜑 → 𝑈 ∈ ℝ) |
ivth.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
ivth.5 | ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) |
ivth.7 | ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) |
ivth.8 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) |
ivth.9 | ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) |
ivth.10 | ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} |
Ref | Expression |
---|---|
ivthlem1 | ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ivth.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 1 | rexrd 9968 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
3 | ivth.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 3 | rexrd 9968 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
5 | ivth.4 | . . . . 5 ⊢ (𝜑 → 𝐴 < 𝐵) | |
6 | 1, 3, 5 | ltled 10064 | . . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
7 | lbicc2 12159 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
8 | 2, 4, 6, 7 | syl3anc 1318 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
9 | ivth.8 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) | |
10 | 9 | ralrimiva 2949 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ) |
11 | fveq2 6103 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
12 | 11 | eleq1d 2672 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐴) ∈ ℝ)) |
13 | 12 | rspcv 3278 | . . . . 5 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → (∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ → (𝐹‘𝐴) ∈ ℝ)) |
14 | 8, 10, 13 | sylc 63 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ) |
15 | ivth.3 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ ℝ) | |
16 | ivth.9 | . . . . 5 ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) | |
17 | 16 | simpld 474 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) < 𝑈) |
18 | 14, 15, 17 | ltled 10064 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ≤ 𝑈) |
19 | 11 | breq1d 4593 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ≤ 𝑈 ↔ (𝐹‘𝐴) ≤ 𝑈)) |
20 | ivth.10 | . . . 4 ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} | |
21 | 19, 20 | elrab2 3333 | . . 3 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝐴) ≤ 𝑈)) |
22 | 8, 18, 21 | sylanbrc 695 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
23 | ssrab2 3650 | . . . . . 6 ⊢ {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} ⊆ (𝐴[,]𝐵) | |
24 | 20, 23 | eqsstri 3598 | . . . . 5 ⊢ 𝑆 ⊆ (𝐴[,]𝐵) |
25 | 24 | sseli 3564 | . . . 4 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ (𝐴[,]𝐵)) |
26 | iccleub 12100 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ≤ 𝐵) | |
27 | 26 | 3expia 1259 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑧 ∈ (𝐴[,]𝐵) → 𝑧 ≤ 𝐵)) |
28 | 2, 4, 27 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴[,]𝐵) → 𝑧 ≤ 𝐵)) |
29 | 25, 28 | syl5 33 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑆 → 𝑧 ≤ 𝐵)) |
30 | 29 | ralrimiv 2948 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵) |
31 | 22, 30 | jca 553 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 [,]cicc 12049 –cn→ccncf 22487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-icc 12053 |
This theorem is referenced by: ivthlem2 23028 ivthlem3 23029 |
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