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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 729 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝐴 ⊆ ℝ) |
3 | ovolshft.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | ad3antrrr 729 | . . . . . . 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 729 | . . . . . . 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 2798 | . . . . . . 7 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔)) | |
9 | 2fveq3 6650 | . . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (1st ‘(𝑔‘𝑚)) = (1st ‘(𝑔‘𝑛))) | |
10 | 9 | oveq1d 7150 | . . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((1st ‘(𝑔‘𝑚)) + 𝐶) = ((1st ‘(𝑔‘𝑛)) + 𝐶)) |
11 | 2fveq3 6650 | . . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝑔‘𝑚)) = (2nd ‘(𝑔‘𝑛))) | |
12 | 11 | oveq1d 7150 | . . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((2nd ‘(𝑔‘𝑚)) + 𝐶) = ((2nd ‘(𝑔‘𝑛)) + 𝐶)) |
13 | 10, 12 | opeq12d 4773 | . . . . . . . 8 ⊢ (𝑚 = 𝑛 → 〈((1st ‘(𝑔‘𝑚)) + 𝐶), ((2nd ‘(𝑔‘𝑚)) + 𝐶)〉 = 〈((1st ‘(𝑔‘𝑛)) + 𝐶), ((2nd ‘(𝑔‘𝑛)) + 𝐶)〉) |
14 | 13 | cbvmptv 5133 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ 〈((1st ‘(𝑔‘𝑚)) + 𝐶), ((2nd ‘(𝑔‘𝑚)) + 𝐶)〉) = (𝑛 ∈ ℕ ↦ 〈((1st ‘(𝑔‘𝑛)) + 𝐶), ((2nd ‘(𝑔‘𝑛)) + 𝐶)〉) |
15 | simplr 768 | . . . . . . . 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 221 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ))) |
18 | simpr 488 | . . . . . . 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 2885 | . . . . . 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 457 | . . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ*) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) → 𝑧 ∈ 𝑀)) |
23 | 22 | rexlimdva 3243 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ*) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) → 𝑧 ∈ 𝑀)) |
24 | 23 | ralrimiva 3149 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ ℝ* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < )) → 𝑧 ∈ 𝑀)) |
25 | rabss 3999 | . 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 237 | 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 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 {crab 3110 ∩ cin 3880 ⊆ wss 3881 〈cop 4531 ∪ cuni 4800 ↦ cmpt 5110 × cxp 5517 ran crn 5520 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1st c1st 7669 2nd c2nd 7670 ↑m cmap 8389 supcsup 8888 ℝcr 10525 1c1 10527 + caddc 10529 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 ℕcn 11625 (,)cioo 12726 seqcseq 13364 abscabs 14585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ioo 12730 df-ico 12732 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: ovolshft 24115 |
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