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| Description: Lemma for ivth 25489. 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 11311 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | 
| 3 | ivth.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 4 | 3 | rexrd 11311 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | 
| 5 | ivth.4 | . . . . 5 ⊢ (𝜑 → 𝐴 < 𝐵) | |
| 6 | 1, 3, 5 | ltled 11409 | . . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | 
| 7 | lbicc2 13504 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
| 8 | 2, 4, 6, 7 | syl3anc 1373 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) | 
| 9 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
| 10 | 9 | eleq1d 2826 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐴) ∈ ℝ)) | 
| 11 | ivth.8 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) | |
| 12 | 11 | ralrimiva 3146 | . . . . 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 11409 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ≤ 𝑈) | 
| 18 | 9 | breq1d 5153 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ≤ 𝑈 ↔ (𝐹‘𝐴) ≤ 𝑈)) | 
| 19 | ivth.10 | . . . 4 ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} | |
| 20 | 18, 19 | elrab2 3695 | . . 3 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝐴) ≤ 𝑈)) | 
| 21 | 8, 17, 20 | sylanbrc 583 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | 
| 22 | 19 | ssrab3 4082 | . . . . 5 ⊢ 𝑆 ⊆ (𝐴[,]𝐵) | 
| 23 | 22 | sseli 3979 | . . . 4 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ (𝐴[,]𝐵)) | 
| 24 | iccleub 13442 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ≤ 𝐵) | |
| 25 | 24 | 3expia 1122 | . . . . 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 3145 | . 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 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 [,]cicc 13390 –cn→ccncf 24902 | 
| 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-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 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 | 
| This theorem is referenced by: ivthlem2 25487 ivthlem3 25488 | 
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