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Mirrors > Home > MPE Home > Th. List > Mathboxes > fct2relem | Structured version Visualization version GIF version |
Description: Lemma for ftc2re 34592. (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 2849 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐶(,)𝐷)) |
4 | eliooxr 13442 | . . . . 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 13443 | . . . . 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 2849 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐶(,)𝐷)) |
13 | eliooord 13443 | . . . . 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 13453 | . . 3 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) ∧ (𝐶 < 𝐴 ∧ 𝐵 < 𝐷)) → (𝐴[,]𝐵) ⊆ (𝐶(,)𝐷)) | |
17 | 6, 7, 10, 15, 16 | syl22anc 839 | . 2 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (𝐶(,)𝐷)) |
18 | 17, 2 | sseqtrrdi 4047 | 1 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 class class class wbr 5148 (class class class)co 7431 ℝ*cxr 11292 < clt 11293 (,)cioo 13384 [,]cicc 13387 |
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-iun 4998 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-1st 8013 df-2nd 8014 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-ioo 13388 df-icc 13391 |
This theorem is referenced by: ftc2re 34592 fdvposlt 34593 fdvneggt 34594 fdvposle 34595 fdvnegge 34596 logdivsqrle 34644 |
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