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Theorem dvle2 42029

Description: Collapsed dvle 26066. (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 2829	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝑅 ∈ ℝ))
3		dvle2.3	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ))
4		cncff 24938	. . . . . . 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 2740	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸)
7	6	fmpt 7144	. . . . . 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 11340	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
11		dvle2.13	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12	9	leidd 11856	. . . . . . 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 11340	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
16		elicc1 13451	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)))
17	15, 10, 16	syl2anc 583	. . . . . 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 3636	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
20		dvle2.8	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐸 = 𝑃)
21	20	eleq1d 2829	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐸 ∈ ℝ ↔ 𝑃 ∈ ℝ))
22	14	leidd 11856	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
23	15, 22, 11	3jca 1128	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))
24		elicc1 13451	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ^ ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
25	15, 10, 24	syl2anc 583	. . . . . 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 3636	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ)
28	19, 27	resubcld 11718	. . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ∈ ℝ)
29		dvle2.11	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐺 = 𝑆)
30	29	eleq1d 2829	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐺 ∈ ℝ ↔ 𝑆 ∈ ℝ))
31		dvle2.4	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ))
32		cncff 24938	. . . . . . 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 2740	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺)
35	34	fmpt 7144	. . . . . 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 3636	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℝ)
38		dvle2.9	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐺 = 𝑄)
39	38	eleq1d 2829	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐺 ∈ ℝ ↔ 𝑄 ∈ ℝ))
40	39, 36, 26	rspcdva 3636	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℝ)
41	37, 40	resubcld 11718	. . 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 26066	. . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))
46		dvle2.12	. . 3 ⊢ (𝜑 → 𝑃 ≤ 𝑄)
47	28, 27, 41, 40, 45, 46	le2addd 11909	. 2 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄))
48	19	recnd 11318	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℂ)
49	27	recnd 11318	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℂ)
50	48, 49	npcand 11651	. . 3 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) = 𝑅)
51	37	recnd 11318	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℂ)
52	40	recnd 11318	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℂ)
53	51, 52	npcand 11651	. . 3 ⊢ (𝜑 → ((𝑆 − 𝑄) + 𝑄) = 𝑆)
54	50, 53	breq12d 5179	. 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 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 class class class wbr 5166 ↦ cmpt 5249 ⟶wf 6569 (class class class)co 7448 ℝcr 11183 + caddc 11187 ℝ^*cxr 11323 ≤ cle 11325 − cmin 11520 (,)cioo 13407 [,]cicc 13410 –cn→ccncf 24921 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: aks4d1p1p5 42032

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