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Mirrors > Home > MPE Home > Th. List > Mathboxes > fct2relem | Structured version Visualization version GIF version |
Description: Lemma for ftc2re 31869. (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 12796 | . . . . 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 12797 | . . . . 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 12797 | . . . . 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 12806 | . . 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 4018 | 1 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 (class class class)co 7156 ℝ*cxr 10674 < clt 10675 (,)cioo 12739 [,]cicc 12742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-ioo 12743 df-icc 12746 |
This theorem is referenced by: ftc2re 31869 fdvposlt 31870 fdvneggt 31871 fdvposle 31872 fdvnegge 31873 logdivsqrle 31921 |
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