|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > fct2relem | Structured version Visualization version GIF version | ||
| Description: Lemma for ftc2re 34614. (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 2850 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐶(,)𝐷)) | 
| 4 | eliooxr 13446 | . . . . 5 ⊢ (𝐴 ∈ (𝐶(,)𝐷) → (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) | |
| 5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) | 
| 6 | 5 | simpld 494 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | 
| 7 | 5 | simprd 495 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ*) | 
| 8 | eliooord 13447 | . . . . 5 ⊢ (𝐴 ∈ (𝐶(,)𝐷) → (𝐶 < 𝐴 ∧ 𝐴 < 𝐷)) | |
| 9 | 3, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 ∧ 𝐴 < 𝐷)) | 
| 10 | 9 | simpld 494 | . . 3 ⊢ (𝜑 → 𝐶 < 𝐴) | 
| 11 | ftc2re.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐸) | |
| 12 | 11, 2 | eleqtrdi 2850 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐶(,)𝐷)) | 
| 13 | eliooord 13447 | . . . . 5 ⊢ (𝐵 ∈ (𝐶(,)𝐷) → (𝐶 < 𝐵 ∧ 𝐵 < 𝐷)) | |
| 14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 < 𝐵 ∧ 𝐵 < 𝐷)) | 
| 15 | 14 | simprd 495 | . . 3 ⊢ (𝜑 → 𝐵 < 𝐷) | 
| 16 | iccssioo 13457 | . . 3 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) ∧ (𝐶 < 𝐴 ∧ 𝐵 < 𝐷)) → (𝐴[,]𝐵) ⊆ (𝐶(,)𝐷)) | |
| 17 | 6, 7, 10, 15, 16 | syl22anc 838 | . 2 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐶(,)𝐷)) | 
| 18 | 17, 2 | sseqtrrdi 4024 | 1 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 class class class wbr 5142 (class class class)co 7432 ℝ*cxr 11295 < clt 11296 (,)cioo 13388 [,]cicc 13391 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-ioo 13392 df-icc 13395 | 
| This theorem is referenced by: ftc2re 34614 fdvposlt 34615 fdvneggt 34616 fdvposle 34617 fdvnegge 34618 logdivsqrle 34666 | 
| Copyright terms: Public domain | W3C validator |