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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvle2 | Structured version Visualization version GIF version |
Description: Collapsed dvle 25171. (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 2823 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝑅 ∈ ℝ)) |
3 | dvle2.3 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
4 | cncff 24056 | . . . . . . 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 6984 | . . . . . 6 ⊢ (∀𝑥 ∈ (𝐴[,]𝐵)𝐸 ∈ ℝ ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸):(𝐴[,]𝐵)⟶ℝ) |
8 | 5, 7 | sylibr 233 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)𝐸 ∈ ℝ) |
9 | dvle2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
10 | 9 | rexrd 11025 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
11 | dvle2.13 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
12 | 9 | leidd 11541 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝐵) |
13 | 10, 11, 12 | 3jca 1127 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)) |
14 | dvle2.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
15 | 14 | rexrd 11025 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
16 | elicc1 13123 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))) | |
17 | 15, 10, 16 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))) |
18 | 13, 17 | mpbird 256 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵)) |
19 | 2, 8, 18 | rspcdva 3562 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ) |
20 | dvle2.8 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐸 = 𝑃) | |
21 | 20 | eleq1d 2823 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝑃 ∈ ℝ)) |
22 | 14 | leidd 11541 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
23 | 15, 22, 11 | 3jca 1127 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) |
24 | elicc1 13123 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) | |
25 | 15, 10, 24 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) |
26 | 23, 25 | mpbird 256 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
27 | 21, 8, 26 | rspcdva 3562 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
28 | 19, 27 | resubcld 11403 | . . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ∈ ℝ) |
29 | dvle2.11 | . . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐺 = 𝑆) | |
30 | 29 | eleq1d 2823 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐺 ∈ ℝ ↔ 𝑆 ∈ ℝ)) |
31 | dvle2.4 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
32 | cncff 24056 | . . . . . . 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 6984 | . . . . . 6 ⊢ (∀𝑥 ∈ (𝐴[,]𝐵)𝐺 ∈ ℝ ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺):(𝐴[,]𝐵)⟶ℝ) |
36 | 33, 35 | sylibr 233 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)𝐺 ∈ ℝ) |
37 | 30, 36, 18 | rspcdva 3562 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
38 | dvle2.9 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐺 = 𝑄) | |
39 | 38 | eleq1d 2823 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐺 ∈ ℝ ↔ 𝑄 ∈ ℝ)) |
40 | 39, 36, 26 | rspcdva 3562 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
41 | 37, 40 | resubcld 11403 | . . 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 25171 | . . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄)) |
46 | dvle2.12 | . . 3 ⊢ (𝜑 → 𝑃 ≤ 𝑄) | |
47 | 28, 27, 41, 40, 45, 46 | le2addd 11594 | . 2 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄)) |
48 | 19 | recnd 11003 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ℂ) |
49 | 27 | recnd 11003 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
50 | 48, 49 | npcand 11336 | . . 3 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) = 𝑅) |
51 | 37 | recnd 11003 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ ℂ) |
52 | 40 | recnd 11003 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
53 | 51, 52 | npcand 11336 | . . 3 ⊢ (𝜑 → ((𝑆 − 𝑄) + 𝑄) = 𝑆) |
54 | 50, 53 | breq12d 5087 | . 2 ⊢ (𝜑 → (((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄) ↔ 𝑅 ≤ 𝑆)) |
55 | 47, 54 | mpbid 231 | 1 ⊢ (𝜑 → 𝑅 ≤ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 class class class wbr 5074 ↦ cmpt 5157 ⟶wf 6429 (class class class)co 7275 ℝcr 10870 + caddc 10874 ℝ*cxr 11008 ≤ cle 11010 − cmin 11205 (,)cioo 13079 [,]cicc 13082 –cn→ccncf 24039 D cdv 25027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 |
This theorem is referenced by: aks4d1p1p5 40083 |
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