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Mirrors > Home > MPE Home > Th. List > Mathboxes > fct2relem | Structured version Visualization version GIF version |
Description: Lemma for ftc2re 31864. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
Ref | Expression |
---|---|
ftc2re.e | ⊢ 𝐸 = (𝐶(,)𝐷) |
ftc2re.a | ⊢ (𝜑 → 𝐴 ∈ 𝐸) |
ftc2re.b | ⊢ (𝜑 → 𝐵 ∈ 𝐸) |
Ref | Expression |
---|---|
fct2relem | ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ftc2re.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐸) | |
2 | ftc2re.e | . . . . . 6 ⊢ 𝐸 = (𝐶(,)𝐷) | |
3 | 1, 2 | eleqtrdi 2923 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐶(,)𝐷)) |
4 | eliooxr 12789 | . . . . 5 ⊢ (𝐴 ∈ (𝐶(,)𝐷) → (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) |
6 | 5 | simpld 497 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
7 | 5 | simprd 498 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ*) |
8 | eliooord 12790 | . . . . 5 ⊢ (𝐴 ∈ (𝐶(,)𝐷) → (𝐶 < 𝐴 ∧ 𝐴 < 𝐷)) | |
9 | 3, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 ∧ 𝐴 < 𝐷)) |
10 | 9 | simpld 497 | . . 3 ⊢ (𝜑 → 𝐶 < 𝐴) |
11 | ftc2re.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐸) | |
12 | 11, 2 | eleqtrdi 2923 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐶(,)𝐷)) |
13 | eliooord 12790 | . . . . 5 ⊢ (𝐵 ∈ (𝐶(,)𝐷) → (𝐶 < 𝐵 ∧ 𝐵 < 𝐷)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 < 𝐵 ∧ 𝐵 < 𝐷)) |
15 | 14 | simprd 498 | . . 3 ⊢ (𝜑 → 𝐵 < 𝐷) |
16 | iccssioo 12799 | . . 3 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) ∧ (𝐶 < 𝐴 ∧ 𝐵 < 𝐷)) → (𝐴[,]𝐵) ⊆ (𝐶(,)𝐷)) | |
17 | 6, 7, 10, 15, 16 | syl22anc 836 | . 2 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐶(,)𝐷)) |
18 | 17, 2 | sseqtrrdi 4017 | 1 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5058 (class class class)co 7150 ℝ*cxr 10668 < clt 10669 (,)cioo 12732 [,]cicc 12735 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-ioo 12736 df-icc 12739 |
This theorem is referenced by: ftc2re 31864 fdvposlt 31865 fdvneggt 31866 fdvposle 31867 fdvnegge 31868 logdivsqrle 31916 |
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