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Theorem dvle2 40008

Description: Collapsed dvle 25076. (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 2823	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐸 ∈ ℝ ↔ 𝑅 ∈ ℝ))
3		dvle2.3	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ))
4		cncff 23962	. . . . . . 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 6966	. . . . . 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 10956	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
11		dvle2.13	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12	9	leidd 11471	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
13	10, 11, 12	3jca 1126	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))
14		dvle2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
15	14	rexrd 10956	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
16		elicc1 13052	. . . . . . 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 256	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
19	2, 8, 18	rspcdva 3554	. . . 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 11471	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
23	15, 22, 11	3jca 1126	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))
24		elicc1 13052	. . . . . . 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 256	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
27	21, 8, 26	rspcdva 3554	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ)
28	19, 27	resubcld 11333	. . 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 23962	. . . . . . 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 6966	. . . . . 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 3554	. . . 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 3554	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℝ)
41	37, 40	resubcld 11333	. . 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 25076	. . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))
46		dvle2.12	. . 3 ⊢ (𝜑 → 𝑃 ≤ 𝑄)
47	28, 27, 41, 40, 45, 46	le2addd 11524	. 2 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄))
48	19	recnd 10934	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℂ)
49	27	recnd 10934	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℂ)
50	48, 49	npcand 11266	. . 3 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) = 𝑅)
51	37	recnd 10934	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℂ)
52	40	recnd 10934	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℂ)
53	51, 52	npcand 11266	. . 3 ⊢ (𝜑 → ((𝑆 − 𝑄) + 𝑄) = 𝑆)
54	50, 53	breq12d 5083	. 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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 class class class wbr 5070 ↦ cmpt 5153 ⟶wf 6414 (class class class)co 7255 ℝcr 10801 + caddc 10805 ℝ^*cxr 10939 ≤ cle 10941 − cmin 11135 (,)cioo 13008 [,]cicc 13011 –cn→ccncf 23945 D cdv 24932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936

This theorem is referenced by: aks4d1p1p5 40011

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