Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ovolshftlem2 | Structured version Visualization version GIF version |
Description: Lemma for ovolshft 24115. (Contributed by Mario Carneiro, 22-Mar-2014.) |
Ref | Expression |
---|---|
ovolshft.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
ovolshft.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ovolshft.3 | ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}) |
ovolshft.4 | ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} |
Ref | Expression |
---|---|
ovolshftlem2 | ⊢ (𝜑 → {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolshft.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | 1 | ad3antrrr 728 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐴 ⊆ ℝ) |
3 | ovolshft.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | ad3antrrr 728 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐶 ∈ ℝ) |
5 | ovolshft.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}) | |
6 | 5 | ad3antrrr 728 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐵 = {𝑥 ∈ ℝ ∣ (𝑥 − 𝐶) ∈ 𝐴}) |
7 | ovolshft.4 | . . . . . . 7 ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} | |
8 | eqid 2824 | . . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔)) | |
9 | 2fveq3 6678 | . . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1st ‘(𝑔‘𝑚)) = (1st ‘(𝑔‘𝑛))) | |
10 | 9 | oveq1d 7174 | . . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((1st ‘(𝑔‘𝑚)) + 𝐶) = ((1st ‘(𝑔‘𝑛)) + 𝐶)) |
11 | 2fveq3 6678 | . . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑔‘𝑚)) = (2nd ‘(𝑔‘𝑛))) | |
12 | 11 | oveq1d 7174 | . . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((2nd ‘(𝑔‘𝑚)) + 𝐶) = ((2nd ‘(𝑔‘𝑛)) + 𝐶)) |
13 | 10, 12 | opeq12d 4814 | . . . . . . . 8 ⊢ (𝑚 = 𝑛 → 〈((1st ‘(𝑔‘𝑚)) + 𝐶), ((2nd ‘(𝑔‘𝑚)) + 𝐶)〉 = 〈((1st ‘(𝑔‘𝑛)) + 𝐶), ((2nd ‘(𝑔‘𝑛)) + 𝐶)〉) |
14 | 13 | cbvmptv 5172 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ 〈((1st ‘(𝑔‘𝑚)) + 𝐶), ((2nd ‘(𝑔‘𝑚)) + 𝐶)〉) = (𝑛 ∈ ℕ ↦ 〈((1st ‘(𝑔‘𝑛)) + 𝐶), ((2nd ‘(𝑔‘𝑛)) + 𝐶)〉) |
15 | simplr 767 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) | |
16 | elovolmlem 24078 | . . . . . . . 8 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ))) | |
17 | 15, 16 | sylib 220 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ))) |
18 | simpr 487 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) | |
19 | 2, 4, 6, 7, 8, 14, 17, 18 | ovolshftlem1 24113 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∈ 𝑀) |
20 | eleq1a 2911 | . . . . . 6 ⊢ (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∈ 𝑀 → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) → 𝑧 ∈ 𝑀)) | |
21 | 19, 20 | syl 17 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) → 𝑧 ∈ 𝑀)) |
22 | 21 | expimpd 456 | . . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) → 𝑧 ∈ 𝑀)) |
23 | 22 | rexlimdva 3287 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ*) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) → 𝑧 ∈ 𝑀)) |
24 | 23 | ralrimiva 3185 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ ℝ* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) → 𝑧 ∈ 𝑀)) |
25 | rabss 4051 | . 2 ⊢ ({𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ 𝑀 ↔ ∀𝑧 ∈ ℝ* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) → 𝑧 ∈ 𝑀)) | |
26 | 24, 25 | sylibr 236 | 1 ⊢ (𝜑 → {𝑧 ∈ ℝ* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))} ⊆ 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 {crab 3145 ∩ cin 3938 ⊆ wss 3939 〈cop 4576 ∪ cuni 4841 ↦ cmpt 5149 × cxp 5556 ran crn 5559 ∘ ccom 5562 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 1st c1st 7690 2nd c2nd 7691 ↑m cmap 8409 supcsup 8907 ℝcr 10539 1c1 10541 + caddc 10543 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 − cmin 10873 ℕcn 11641 (,)cioo 12741 seqcseq 13372 abscabs 14596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ioo 12745 df-ico 12747 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 |
This theorem is referenced by: ovolshft 24115 |
Copyright terms: Public domain | W3C validator |