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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvle2 | Structured version Visualization version GIF version |
Description: Collapsed dvle 26061. (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 2824 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝑅 ∈ ℝ)) |
3 | dvle2.3 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
4 | cncff 24933 | . . . . . . 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 2735 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) | |
7 | 6 | fmpt 7130 | . . . . . 6 ⊢ (∀𝑥 ∈ (𝐴[,]𝐵)𝐸 ∈ ℝ ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸):(𝐴[,]𝐵)⟶ℝ) |
8 | 5, 7 | sylibr 234 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)𝐸 ∈ ℝ) |
9 | dvle2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
10 | 9 | rexrd 11309 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
11 | dvle2.13 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
12 | 9 | leidd 11827 | . . . . . . 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 11309 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
16 | elicc1 13428 | . . . . . . 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 257 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵)) |
19 | 2, 8, 18 | rspcdva 3623 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ) |
20 | dvle2.8 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐸 = 𝑃) | |
21 | 20 | eleq1d 2824 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝑃 ∈ ℝ)) |
22 | 14 | leidd 11827 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
23 | 15, 22, 11 | 3jca 1127 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) |
24 | elicc1 13428 | . . . . . . 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 257 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
27 | 21, 8, 26 | rspcdva 3623 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ) |
28 | 19, 27 | resubcld 11689 | . . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ∈ ℝ) |
29 | dvle2.11 | . . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐺 = 𝑆) | |
30 | 29 | eleq1d 2824 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐺 ∈ ℝ ↔ 𝑆 ∈ ℝ)) |
31 | dvle2.4 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
32 | cncff 24933 | . . . . . . 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 2735 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) | |
35 | 34 | fmpt 7130 | . . . . . 6 ⊢ (∀𝑥 ∈ (𝐴[,]𝐵)𝐺 ∈ ℝ ↔ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺):(𝐴[,]𝐵)⟶ℝ) |
36 | 33, 35 | sylibr 234 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)𝐺 ∈ ℝ) |
37 | 30, 36, 18 | rspcdva 3623 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
38 | dvle2.9 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐺 = 𝑄) | |
39 | 38 | eleq1d 2824 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐺 ∈ ℝ ↔ 𝑄 ∈ ℝ)) |
40 | 39, 36, 26 | rspcdva 3623 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ ℝ) |
41 | 37, 40 | resubcld 11689 | . . 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 26061 | . . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄)) |
46 | dvle2.12 | . . 3 ⊢ (𝜑 → 𝑃 ≤ 𝑄) | |
47 | 28, 27, 41, 40, 45, 46 | le2addd 11880 | . 2 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄)) |
48 | 19 | recnd 11287 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ ℂ) |
49 | 27 | recnd 11287 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ ℂ) |
50 | 48, 49 | npcand 11622 | . . 3 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) = 𝑅) |
51 | 37 | recnd 11287 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ ℂ) |
52 | 40 | recnd 11287 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
53 | 51, 52 | npcand 11622 | . . 3 ⊢ (𝜑 → ((𝑆 − 𝑄) + 𝑄) = 𝑆) |
54 | 50, 53 | breq12d 5161 | . 2 ⊢ (𝜑 → (((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄) ↔ 𝑅 ≤ 𝑆)) |
55 | 47, 54 | mpbid 232 | 1 ⊢ (𝜑 → 𝑅 ≤ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6559 (class class class)co 7431 ℝcr 11152 + caddc 11156 ℝ*cxr 11292 ≤ cle 11294 − cmin 11490 (,)cioo 13384 [,]cicc 13387 –cn→ccncf 24916 D cdv 25913 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
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-rmo 3378 df-reu 3379 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-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 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-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 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-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 |
This theorem is referenced by: aks4d1p1p5 42057 |
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