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Mirrors > Home > MPE Home > Th. List > Mathboxes > sxbrsigalem5 | Structured version Visualization version GIF version |
Description: First direction for sxbrsiga 31658. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.) |
Ref | Expression |
---|---|
sxbrsiga.0 | ⊢ 𝐽 = (topGen‘ran (,)) |
dya2ioc.1 | ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) |
dya2ioc.2 | ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) |
Ref | Expression |
---|---|
sxbrsigalem5 | ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅ℝ ×s 𝔅ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sxbrsiga.0 | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
2 | dya2ioc.1 | . . . . 5 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) | |
3 | dya2ioc.2 | . . . . 5 ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) | |
4 | 1, 2, 3 | dya2iocucvr 31652 | . . . 4 ⊢ ∪ ran 𝑅 = (ℝ × ℝ) |
5 | br2base 31637 | . . . 4 ⊢ ∪ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = (ℝ × ℝ) | |
6 | 4, 5 | eqtr4i 2824 | . . 3 ⊢ ∪ ran 𝑅 = ∪ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) |
7 | brsigarn 31553 | . . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
8 | 7 | elexi 3460 | . . . . . 6 ⊢ 𝔅ℝ ∈ V |
9 | 8, 8 | mpoex 7760 | . . . . 5 ⊢ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) ∈ V |
10 | 9 | rnex 7599 | . . . 4 ⊢ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) ∈ V |
11 | 1, 2 | dya2icobrsiga 31644 | . . . . . . . . . 10 ⊢ ran 𝐼 ⊆ 𝔅ℝ |
12 | 11 | sseli 3911 | . . . . . . . . 9 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ∈ 𝔅ℝ) |
13 | 11 | sseli 3911 | . . . . . . . . 9 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ∈ 𝔅ℝ) |
14 | 12, 13 | anim12i 615 | . . . . . . . 8 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ)) |
15 | 14 | anim1i 617 | . . . . . . 7 ⊢ (((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑔 = (𝑢 × 𝑣)) → ((𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ) ∧ 𝑔 = (𝑢 × 𝑣))) |
16 | 15 | ssoprab2i 7242 | . . . . . 6 ⊢ {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑔 = (𝑢 × 𝑣))} ⊆ {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ) ∧ 𝑔 = (𝑢 × 𝑣))} |
17 | df-mpo 7140 | . . . . . . 7 ⊢ (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣)) = {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑔 = (𝑢 × 𝑣))} | |
18 | 3, 17 | eqtri 2821 | . . . . . 6 ⊢ 𝑅 = {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑔 = (𝑢 × 𝑣))} |
19 | xpeq1 5533 | . . . . . . . 8 ⊢ (𝑒 = 𝑢 → (𝑒 × 𝑓) = (𝑢 × 𝑓)) | |
20 | xpeq2 5540 | . . . . . . . 8 ⊢ (𝑓 = 𝑣 → (𝑢 × 𝑓) = (𝑢 × 𝑣)) | |
21 | 19, 20 | cbvmpov 7228 | . . . . . . 7 ⊢ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = (𝑢 ∈ 𝔅ℝ, 𝑣 ∈ 𝔅ℝ ↦ (𝑢 × 𝑣)) |
22 | df-mpo 7140 | . . . . . . 7 ⊢ (𝑢 ∈ 𝔅ℝ, 𝑣 ∈ 𝔅ℝ ↦ (𝑢 × 𝑣)) = {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ) ∧ 𝑔 = (𝑢 × 𝑣))} | |
23 | 21, 22 | eqtri 2821 | . . . . . 6 ⊢ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ) ∧ 𝑔 = (𝑢 × 𝑣))} |
24 | 16, 18, 23 | 3sstr4i 3958 | . . . . 5 ⊢ 𝑅 ⊆ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) |
25 | rnss 5773 | . . . . 5 ⊢ (𝑅 ⊆ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) → ran 𝑅 ⊆ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) | |
26 | 24, 25 | ax-mp 5 | . . . 4 ⊢ ran 𝑅 ⊆ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) |
27 | sssigagen2 31515 | . . . 4 ⊢ ((ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) ∈ V ∧ ran 𝑅 ⊆ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) → ran 𝑅 ⊆ (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)))) | |
28 | 10, 26, 27 | mp2an 691 | . . 3 ⊢ ran 𝑅 ⊆ (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) |
29 | sigagenss2 31519 | . . 3 ⊢ ((∪ ran 𝑅 = ∪ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) ∧ ran 𝑅 ⊆ (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) ∧ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) ∈ V) → (sigaGen‘ran 𝑅) ⊆ (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)))) | |
30 | 6, 28, 10, 29 | mp3an 1458 | . 2 ⊢ (sigaGen‘ran 𝑅) ⊆ (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) |
31 | 1, 2, 3 | sxbrsigalem4 31655 | . 2 ⊢ (sigaGen‘(𝐽 ×t 𝐽)) = (sigaGen‘ran 𝑅) |
32 | eqid 2798 | . . . 4 ⊢ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) | |
33 | 32 | sxval 31559 | . . 3 ⊢ ((𝔅ℝ ∈ (sigAlgebra‘ℝ) ∧ 𝔅ℝ ∈ (sigAlgebra‘ℝ)) → (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)))) |
34 | 7, 7, 33 | mp2an 691 | . 2 ⊢ (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) |
35 | 30, 31, 34 | 3sstr4i 3958 | 1 ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅ℝ ×s 𝔅ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ∪ cuni 4800 × cxp 5517 ran crn 5520 ‘cfv 6324 (class class class)co 7135 {coprab 7136 ∈ cmpo 7137 ℝcr 10525 1c1 10527 + caddc 10529 / cdiv 11286 2c2 11680 ℤcz 11969 (,)cioo 12726 [,)cico 12728 ↑cexp 13425 topGenctg 16703 ×t ctx 22165 sigAlgebracsiga 31477 sigaGencsigagen 31507 𝔅ℝcbrsiga 31550 ×s csx 31557 |
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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-ac2 9874 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 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 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-int 4839 df-iun 4883 df-iin 4884 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-se 5479 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-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-acn 9355 df-ac 9527 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-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-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-refld 20294 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-fcls 22546 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-cfil 23859 df-cmet 23861 df-cms 23939 df-limc 24469 df-dv 24470 df-log 25148 df-cxp 25149 df-logb 25351 df-siga 31478 df-sigagen 31508 df-brsiga 31551 df-sx 31558 |
This theorem is referenced by: sxbrsigalem6 31657 |
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