dvivthlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvivthlem2		Structured version Visualization version GIF version

Theorem dvivthlem2 26049

Description: Lemma for dvivth 26050. (Contributed by Mario Carneiro, 20-Feb-2015.)

**Hypotheses**
Ref	Expression
dvivth.1	⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵))
dvivth.2	⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵))
dvivth.3	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
dvivth.4	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
dvivth.5	⊢ (𝜑 → 𝑀 < 𝑁)
dvivth.6	⊢ (𝜑 → 𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)))
dvivth.7	⊢ 𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦)))

**Assertion**
Ref	Expression
dvivthlem2	⊢ (𝜑 → 𝐶 ∈ ran (ℝ D 𝐹))

Distinct variable groups: 𝑦,𝐴 𝑦,𝐵 𝑦,𝐹 𝑦,𝑀 𝑦,𝐶 𝑦,𝑁 𝜑,𝑦 Allowed substitution hint: 𝐺(𝑦)

Proof of Theorem dvivthlem2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvivth.1	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵))
2		dvivth.2	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵))
3		dvivth.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
4		dvivth.4	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
5		dvivth.5	. . 3 ⊢ (𝜑 → 𝑀 < 𝑁)
6		dvivth.6	. . 3 ⊢ (𝜑 → 𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)))
7		dvivth.7	. . 3 ⊢ 𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦)))
8	1, 2, 3, 4, 5, 6, 7	dvivthlem1 26048	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶)
9		dvf 25943	. . . . . . 7 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
10	4	feq2d 6721	. . . . . . 7 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
11	9, 10	mpbii 233	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
12	11	ffnd 6736	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
13		iccssioo2 13461	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐴(,)𝐵) ∧ 𝑁 ∈ (𝐴(,)𝐵)) → (𝑀[,]𝑁) ⊆ (𝐴(,)𝐵))
14	1, 2, 13	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ (𝐴(,)𝐵))
15	14	sselda 3982	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝑥 ∈ (𝐴(,)𝐵))
16		fnfvelrn 7099	. . . . 5 ⊢ (((ℝ D 𝐹) Fn (𝐴(,)𝐵) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ran (ℝ D 𝐹))
17	12, 15, 16	syl2an2r 685	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐹)‘𝑥) ∈ ran (ℝ D 𝐹))
18		eleq1 2828	. . . 4 ⊢ (((ℝ D 𝐹)‘𝑥) = 𝐶 → (((ℝ D 𝐹)‘𝑥) ∈ ran (ℝ D 𝐹) ↔ 𝐶 ∈ ran (ℝ D 𝐹)))
19	17, 18	syl5ibcom 245	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (((ℝ D 𝐹)‘𝑥) = 𝐶 → 𝐶 ∈ ran (ℝ D 𝐹)))
20	19	rexlimdva 3154	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶 → 𝐶 ∈ ran (ℝ D 𝐹)))
21	8, 20	mpd 15	1 ⊢ (𝜑 → 𝐶 ∈ ran (ℝ D 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ⊆ wss 3950 class class class wbr 5142 ↦ cmpt 5224 dom cdm 5684 ran crn 5685 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 · cmul 11161 < clt 11296 − cmin 11493 (,)cioo 13388 [,]cicc 13391 –cn→ccncf 24903 D cdv 25899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903

This theorem is referenced by: dvivth 26050

W3C validator