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Mirrors > Home > MPE Home > Th. List > ivthlem1 | Structured version Visualization version GIF version |
Description: Lemma for ivth 25503. 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 11309 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
3 | ivth.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 3 | rexrd 11309 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
5 | ivth.4 | . . . . 5 ⊢ (𝜑 → 𝐴 < 𝐵) | |
6 | 1, 3, 5 | ltled 11407 | . . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
7 | lbicc2 13501 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
8 | 2, 4, 6, 7 | syl3anc 1370 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
9 | fveq2 6907 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
10 | 9 | eleq1d 2824 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐴) ∈ ℝ)) |
11 | ivth.8 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) | |
12 | 11 | ralrimiva 3144 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ) |
13 | 10, 12, 8 | rspcdva 3623 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ) |
14 | ivth.3 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ ℝ) | |
15 | ivth.9 | . . . . 5 ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) | |
16 | 15 | simpld 494 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) < 𝑈) |
17 | 13, 14, 16 | ltled 11407 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ≤ 𝑈) |
18 | 9 | breq1d 5158 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ≤ 𝑈 ↔ (𝐹‘𝐴) ≤ 𝑈)) |
19 | ivth.10 | . . . 4 ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} | |
20 | 18, 19 | elrab2 3698 | . . 3 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝐴) ≤ 𝑈)) |
21 | 8, 17, 20 | sylanbrc 583 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
22 | 19 | ssrab3 4092 | . . . . 5 ⊢ 𝑆 ⊆ (𝐴[,]𝐵) |
23 | 22 | sseli 3991 | . . . 4 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ (𝐴[,]𝐵)) |
24 | iccleub 13439 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ≤ 𝐵) | |
25 | 24 | 3expia 1120 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑧 ∈ (𝐴[,]𝐵) → 𝑧 ≤ 𝐵)) |
26 | 2, 4, 25 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴[,]𝐵) → 𝑧 ≤ 𝐵)) |
27 | 23, 26 | syl5 34 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑆 → 𝑧 ≤ 𝐵)) |
28 | 27 | ralrimiv 3143 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵) |
29 | 21, 28 | jca 511 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ⊆ wss 3963 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 [,]cicc 13387 –cn→ccncf 24916 |
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-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 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 |
This theorem is referenced by: ivthlem2 25501 ivthlem3 25502 |
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