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Theorem dvle2 40925

Description: Collapsed dvle 25515. (Contributed by metakunt, 19-Aug-2024.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
dvle2	⊢ (𝜑 → 𝑅 ≤ 𝑆)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝑃 𝑥,𝑄 𝑥,𝑅 𝑥,𝑆 𝜑,𝑥 Allowed substitution hints: 𝐸(𝑥) 𝐹(𝑥) 𝐺(𝑥) 𝐻(𝑥)

**Proof of Theorem dvle2**
Step	Hyp	Ref	Expression
1		dvle2.10	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐸 = 𝑅)
2	1	eleq1d 2818	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝑅 ∈ ℝ))
3		dvle2.3	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ))
4		cncff 24400	. . . . . . 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 2732	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸)
7	6	fmpt 7106	. . . . . 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 11260	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
11		dvle2.13	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12	9	leidd 11776	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
13	10, 11, 12	3jca 1128	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))
14		dvle2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
15	14	rexrd 11260	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
16		elicc1 13364	. . . . . . 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 3613	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
20		dvle2.8	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐸 = 𝑃)
21	20	eleq1d 2818	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝑃 ∈ ℝ))
22	14	leidd 11776	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
23	15, 22, 11	3jca 1128	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))
24		elicc1 13364	. . . . . . 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 3613	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ)
28	19, 27	resubcld 11638	. . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ∈ ℝ)
29		dvle2.11	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐺 = 𝑆)
30	29	eleq1d 2818	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐺 ∈ ℝ ↔ 𝑆 ∈ ℝ))
31		dvle2.4	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ))
32		cncff 24400	. . . . . . 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 2732	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺)
35	34	fmpt 7106	. . . . . 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 3613	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℝ)
38		dvle2.9	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐺 = 𝑄)
39	38	eleq1d 2818	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐺 ∈ ℝ ↔ 𝑄 ∈ ℝ))
40	39, 36, 26	rspcdva 3613	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℝ)
41	37, 40	resubcld 11638	. . 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 25515	. . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))
46		dvle2.12	. . 3 ⊢ (𝜑 → 𝑃 ≤ 𝑄)
47	28, 27, 41, 40, 45, 46	le2addd 11829	. 2 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄))
48	19	recnd 11238	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℂ)
49	27	recnd 11238	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℂ)
50	48, 49	npcand 11571	. . 3 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) = 𝑅)
51	37	recnd 11238	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℂ)
52	40	recnd 11238	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℂ)
53	51, 52	npcand 11571	. . 3 ⊢ (𝜑 → ((𝑆 − 𝑄) + 𝑄) = 𝑆)
54	50, 53	breq12d 5160	. 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 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6536 (class class class)co 7405 ℝcr 11105 + caddc 11109 ℝ^*cxr 11243 ≤ cle 11245 − cmin 11440 (,)cioo 13320 [,]cicc 13323 –cn→ccncf 24383 D cdv 25371

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375

This theorem is referenced by: aks4d1p1p5 40928

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