Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dvivthlem2 | Structured version Visualization version GIF version |
Description: Lemma for dvivth 24534. (Contributed by Mario Carneiro, 20-Feb-2015.) |
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))) |
Ref | Expression |
---|---|
dvivthlem2 | ⊢ (𝜑 → 𝐶 ∈ ran (ℝ D 𝐹)) |
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 24532 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶) |
9 | dvf 24432 | . . . . . . 7 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ | |
10 | 4 | feq2d 6493 | . . . . . . 7 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)) |
11 | 9, 10 | mpbii 234 | . . . . . 6 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ) |
12 | 11 | ffnd 6508 | . . . . 5 ⊢ (𝜑 → (ℝ D 𝐹) Fn (𝐴(,)𝐵)) |
13 | iccssioo2 12797 | . . . . . . 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 3964 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝑥 ∈ (𝐴(,)𝐵)) |
16 | fnfvelrn 6840 | . . . . 5 ⊢ (((ℝ D 𝐹) Fn (𝐴(,)𝐵) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ran (ℝ D 𝐹)) | |
17 | 12, 15, 16 | syl2an2r 681 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐹)‘𝑥) ∈ ran (ℝ D 𝐹)) |
18 | eleq1 2897 | . . . 4 ⊢ (((ℝ D 𝐹)‘𝑥) = 𝐶 → (((ℝ D 𝐹)‘𝑥) ∈ ran (ℝ D 𝐹) ↔ 𝐶 ∈ ran (ℝ D 𝐹))) | |
19 | 17, 18 | syl5ibcom 246 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (((ℝ D 𝐹)‘𝑥) = 𝐶 → 𝐶 ∈ ran (ℝ D 𝐹))) |
20 | 19 | rexlimdva 3281 | . 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 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ran crn 5549 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 · cmul 10530 < clt 10663 − cmin 10858 (,)cioo 12726 [,]cicc 12729 –cn→ccncf 23411 D cdv 24388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-cmp 21923 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 |
This theorem is referenced by: dvivth 24534 |
Copyright terms: Public domain | W3C validator |