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Theorem dvle2 42729

Description: Collapsed dvle 26135. (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 2854	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝑅 ∈ ℝ))
3		dvle2.3	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ))
4		cncff 25021	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸):(𝐴[,]𝐵)⟶ℝ)
5	3, 4	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸):(𝐴[,]𝐵)⟶ℝ)
6		eqid 2769	. . . . . . 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 237	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)𝐸 ∈ ℝ)
9		dvle2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
10	9	rexrd 11259	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
11		dvle2.13	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12	9	leidd 11780	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
13	10, 11, 12	3jca 1144	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))
14		dvle2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
15	14	rexrd 11259	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
16		elicc1 13416	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)))
17	15, 10, 16	syl2anc 595	. . . . . 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 3591	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
20		dvle2.8	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐸 = 𝑃)
21	20	eleq1d 2854	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝑃 ∈ ℝ))
22	14	leidd 11780	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
23	15, 22, 11	3jca 1144	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))
24		elicc1 13416	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ^ ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
25	15, 10, 24	syl2anc 595	. . . . . 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 3591	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ)
28	19, 27	resubcld 11642	. . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ∈ ℝ)
29		dvle2.11	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐺 = 𝑆)
30	29	eleq1d 2854	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐺 ∈ ℝ ↔ 𝑆 ∈ ℝ))
31		dvle2.4	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ))
32		cncff 25021	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺):(𝐴[,]𝐵)⟶ℝ)
33	31, 32	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺):(𝐴[,]𝐵)⟶ℝ)
34		eqid 2769	. . . . . . 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 237	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)𝐺 ∈ ℝ)
37	30, 36, 18	rspcdva 3591	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℝ)
38		dvle2.9	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐺 = 𝑄)
39	38	eleq1d 2854	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐺 ∈ ℝ ↔ 𝑄 ∈ ℝ))
40	39, 36, 26	rspcdva 3591	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℝ)
41	37, 40	resubcld 11642	. . 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 26135	. . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))
46		dvle2.12	. . 3 ⊢ (𝜑 → 𝑃 ≤ 𝑄)
47	28, 27, 41, 40, 45, 46	le2addd 11833	. 2 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄))
48	19	recnd 11237	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℂ)
49	27	recnd 11237	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℂ)
50	48, 49	npcand 11573	. . 3 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) = 𝑅)
51	37	recnd 11237	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℂ)
52	40	recnd 11237	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℂ)
53	51, 52	npcand 11573	. . 3 ⊢ (𝜑 → ((𝑆 − 𝑄) + 𝑄) = 𝑆)
54	50, 53	breq12d 5126	. 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 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 class class class wbr 5113 ↦ cmpt 5196 ⟶wf 6533 (class class class)co 7411 ℝcr 11099 + caddc 11103 ℝ^*cxr 11242 ≤ cle 11244 − cmin 11441 (,)cioo 13372 [,]cicc 13375 –cn→ccncf 25004 D cdv 25991

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-lp 23262 df-perf 23263 df-cn 23353 df-cnp 23354 df-haus 23441 df-cmp 23513 df-tx 23688 df-hmeo 23881 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-xms 24446 df-ms 24447 df-tms 24448 df-cncf 25006 df-limc 25994 df-dv 25995

This theorem is referenced by: aks4d1p1p5 42732

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