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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvle2 | Structured version Visualization version GIF version |
Description: Collapsed dvle 24751. (Contributed by metakunt, 19-Aug-2024.) |
Ref | Expression |
---|---|
dvle2.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dvle2.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
dvle2.3 | ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
dvle2.4 | ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
dvle2.5 | ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐸)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐹)) |
dvle2.6 | ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐺)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐻)) |
dvle2.7 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐹 ≤ 𝐻) |
dvle2.8 | ⊢ (𝑥 = 𝐴 → 𝐸 = 𝑃) |
dvle2.9 | ⊢ (𝑥 = 𝐴 → 𝐺 = 𝑄) |
dvle2.10 | ⊢ (𝑥 = 𝐵 → 𝐸 = 𝑅) |
dvle2.11 | ⊢ (𝑥 = 𝐵 → 𝐺 = 𝑆) |
dvle2.12 | ⊢ (𝜑 → 𝑃 ≤ 𝑄) |
dvle2.13 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Ref | Expression |
---|---|
dvle2 | ⊢ (𝜑 → 𝑅 ≤ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvle2.10 | . . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐸 = 𝑅) | |
2 | 1 | eleq1d 2817 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝑅 ∈ ℝ)) |
3 | dvle2.3 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
4 | cncff 23638 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸):(𝐴[,]𝐵)⟶ℝ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸):(𝐴[,]𝐵)⟶ℝ) |
6 | eqid 2738 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) | |
7 | 6 | fmpt 6878 | . . . . . 6 ⊢ (∀𝑥 ∈ (𝐴[,]𝐵)𝐸 ∈ ℝ ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸):(𝐴[,]𝐵)⟶ℝ) |
8 | 5, 7 | sylibr 237 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)𝐸 ∈ ℝ) |
9 | dvle2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
10 | 9 | rexrd 10762 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
11 | dvle2.13 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
12 | 9 | leidd 11277 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝐵) |
13 | 10, 11, 12 | 3jca 1129 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)) |
14 | dvle2.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
15 | 14 | rexrd 10762 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
16 | elicc1 12858 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))) | |
17 | 15, 10, 16 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))) |
18 | 13, 17 | mpbird 260 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵)) |
19 | 2, 8, 18 | rspcdva 3526 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ) |
20 | dvle2.8 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐸 = 𝑃) | |
21 | 20 | eleq1d 2817 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝑃 ∈ ℝ)) |
22 | 14 | leidd 11277 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
23 | 15, 22, 11 | 3jca 1129 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) |
24 | elicc1 12858 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) | |
25 | 15, 10, 24 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) |
26 | 23, 25 | mpbird 260 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
27 | 21, 8, 26 | rspcdva 3526 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
28 | 19, 27 | resubcld 11139 | . . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ∈ ℝ) |
29 | dvle2.11 | . . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐺 = 𝑆) | |
30 | 29 | eleq1d 2817 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐺 ∈ ℝ ↔ 𝑆 ∈ ℝ)) |
31 | dvle2.4 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
32 | cncff 23638 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺):(𝐴[,]𝐵)⟶ℝ) | |
33 | 31, 32 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺):(𝐴[,]𝐵)⟶ℝ) |
34 | eqid 2738 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) | |
35 | 34 | fmpt 6878 | . . . . . 6 ⊢ (∀𝑥 ∈ (𝐴[,]𝐵)𝐺 ∈ ℝ ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺):(𝐴[,]𝐵)⟶ℝ) |
36 | 33, 35 | sylibr 237 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)𝐺 ∈ ℝ) |
37 | 30, 36, 18 | rspcdva 3526 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
38 | dvle2.9 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐺 = 𝑄) | |
39 | 38 | eleq1d 2817 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐺 ∈ ℝ ↔ 𝑄 ∈ ℝ)) |
40 | 39, 36, 26 | rspcdva 3526 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
41 | 37, 40 | resubcld 11139 | . . 3 ⊢ (𝜑 → (𝑆 − 𝑄) ∈ ℝ) |
42 | dvle2.5 | . . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐸)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐹)) | |
43 | dvle2.6 | . . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐺)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐻)) | |
44 | dvle2.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐹 ≤ 𝐻) | |
45 | 14, 9, 3, 42, 31, 43, 44, 26, 18, 11, 20, 38, 1, 29 | dvle 24751 | . . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄)) |
46 | dvle2.12 | . . 3 ⊢ (𝜑 → 𝑃 ≤ 𝑄) | |
47 | 28, 27, 41, 40, 45, 46 | le2addd 11330 | . 2 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄)) |
48 | 19 | recnd 10740 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ℂ) |
49 | 27 | recnd 10740 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
50 | 48, 49 | npcand 11072 | . . 3 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) = 𝑅) |
51 | 37 | recnd 10740 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ ℂ) |
52 | 40 | recnd 10740 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
53 | 51, 52 | npcand 11072 | . . 3 ⊢ (𝜑 → ((𝑆 − 𝑄) + 𝑄) = 𝑆) |
54 | 50, 53 | breq12d 5040 | . 2 ⊢ (𝜑 → (((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄) ↔ 𝑅 ≤ 𝑆)) |
55 | 47, 54 | mpbid 235 | 1 ⊢ (𝜑 → 𝑅 ≤ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ∀wral 3053 class class class wbr 5027 ↦ cmpt 5107 ⟶wf 6329 (class class class)co 7164 ℝcr 10607 + caddc 10611 ℝ*cxr 10745 ≤ cle 10747 − cmin 10941 (,)cioo 12814 [,]cicc 12817 –cn→ccncf 23621 D cdv 24607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 ax-addf 10687 ax-mulf 10688 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-er 8313 df-map 8432 df-pm 8433 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-fi 8941 df-sup 8972 df-inf 8973 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-ioo 12818 df-ico 12820 df-icc 12821 df-fz 12975 df-fzo 13118 df-seq 13454 df-exp 13515 df-hash 13776 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-hom 16685 df-cco 16686 df-rest 16792 df-topn 16793 df-0g 16811 df-gsum 16812 df-topgen 16813 df-pt 16814 df-prds 16817 df-xrs 16871 df-qtop 16876 df-imas 16877 df-xps 16879 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-mulg 18336 df-cntz 18558 df-cmn 19019 df-psmet 20202 df-xmet 20203 df-met 20204 df-bl 20205 df-mopn 20206 df-fbas 20207 df-fg 20208 df-cnfld 20211 df-top 21638 df-topon 21655 df-topsp 21677 df-bases 21690 df-cld 21763 df-ntr 21764 df-cls 21765 df-nei 21842 df-lp 21880 df-perf 21881 df-cn 21971 df-cnp 21972 df-haus 22059 df-cmp 22131 df-tx 22306 df-hmeo 22499 df-fil 22590 df-fm 22682 df-flim 22683 df-flf 22684 df-xms 23066 df-ms 23067 df-tms 23068 df-cncf 23623 df-limc 24610 df-dv 24611 |
This theorem is referenced by: aks4d1p1p5 39691 |
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