dvle2 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvle2		Structured version Visualization version GIF version

Theorem dvle2 42185

Description: Collapsed dvle 25940. (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 24814	. . . . . . 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 2733	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸)
7	6	fmpt 7049	. . . . . 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 11169	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
11		dvle2.13	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12	9	leidd 11690	. . . . . . 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 11169	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
16		elicc1 13291	. . . . . . 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 3574	. . . 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 11690	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
23	15, 22, 11	3jca 1128	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))
24		elicc1 13291	. . . . . . 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 3574	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ)
28	19, 27	resubcld 11552	. . 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 24814	. . . . . . 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 2733	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺)
35	34	fmpt 7049	. . . . . 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 3574	. . . 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 3574	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℝ)
41	37, 40	resubcld 11552	. . 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 25940	. . 3 ⊢ (𝜑 → (𝑅 − 𝑃) ≤ (𝑆 − 𝑄))
46		dvle2.12	. . 3 ⊢ (𝜑 → 𝑃 ≤ 𝑄)
47	28, 27, 41, 40, 45, 46	le2addd 11743	. 2 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) ≤ ((𝑆 − 𝑄) + 𝑄))
48	19	recnd 11147	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℂ)
49	27	recnd 11147	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℂ)
50	48, 49	npcand 11483	. . 3 ⊢ (𝜑 → ((𝑅 − 𝑃) + 𝑃) = 𝑅)
51	37	recnd 11147	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℂ)
52	40	recnd 11147	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ℂ)
53	51, 52	npcand 11483	. . 3 ⊢ (𝜑 → ((𝑆 − 𝑄) + 𝑄) = 𝑆)
54	50, 53	breq12d 5106	. 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 1541 ∈ wcel 2113 ∀wral 3048 class class class wbr 5093 ↦ cmpt 5174 ⟶wf 6482 (class class class)co 7352 ℝcr 11012 + caddc 11016 ℝ^*cxr 11152 ≤ cle 11154 − cmin 11351 (,)cioo 13247 [,]cicc 13250 –cn→ccncf 24797 D cdv 25792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091 ax-addf 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-fi 9302 df-sup 9333 df-inf 9334 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-ioo 13251 df-ico 13253 df-icc 13254 df-fz 13410 df-fzo 13557 df-seq 13911 df-exp 13971 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-rest 17328 df-topn 17329 df-0g 17347 df-gsum 17348 df-topgen 17349 df-pt 17350 df-prds 17353 df-xrs 17408 df-qtop 17413 df-imas 17414 df-xps 17416 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19231 df-cmn 19696 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-lp 23052 df-perf 23053 df-cn 23143 df-cnp 23144 df-haus 23231 df-cmp 23303 df-tx 23478 df-hmeo 23671 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-tms 24238 df-cncf 24799 df-limc 25795 df-dv 25796

This theorem is referenced by: aks4d1p1p5 42188

W3C validator