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Mirrors > Home > MPE Home > Th. List > Mathboxes > sxbrsigalem5 | Structured version Visualization version GIF version |
Description: First direction for sxbrsiga 31550. (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 31544 | . . . 4 ⊢ ∪ ran 𝑅 = (ℝ × ℝ) |
5 | br2base 31529 | . . . 4 ⊢ ∪ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = (ℝ × ℝ) | |
6 | 4, 5 | eqtr4i 2849 | . . 3 ⊢ ∪ ran 𝑅 = ∪ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) |
7 | brsigarn 31445 | . . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
8 | 7 | elexi 3515 | . . . . . 6 ⊢ 𝔅ℝ ∈ V |
9 | 8, 8 | mpoex 7779 | . . . . 5 ⊢ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) ∈ V |
10 | 9 | rnex 7619 | . . . 4 ⊢ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) ∈ V |
11 | 1, 2 | dya2icobrsiga 31536 | . . . . . . . . . 10 ⊢ ran 𝐼 ⊆ 𝔅ℝ |
12 | 11 | sseli 3965 | . . . . . . . . 9 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ∈ 𝔅ℝ) |
13 | 11 | sseli 3965 | . . . . . . . . 9 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ∈ 𝔅ℝ) |
14 | 12, 13 | anim12i 614 | . . . . . . . 8 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ)) |
15 | 14 | anim1i 616 | . . . . . . 7 ⊢ (((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑔 = (𝑢 × 𝑣)) → ((𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ) ∧ 𝑔 = (𝑢 × 𝑣))) |
16 | 15 | ssoprab2i 7265 | . . . . . 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 7163 | . . . . . . 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 2846 | . . . . . 6 ⊢ 𝑅 = {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑔 = (𝑢 × 𝑣))} |
19 | xpeq1 5571 | . . . . . . . 8 ⊢ (𝑒 = 𝑢 → (𝑒 × 𝑓) = (𝑢 × 𝑓)) | |
20 | xpeq2 5578 | . . . . . . . 8 ⊢ (𝑓 = 𝑣 → (𝑢 × 𝑓) = (𝑢 × 𝑣)) | |
21 | 19, 20 | cbvmpov 7251 | . . . . . . 7 ⊢ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = (𝑢 ∈ 𝔅ℝ, 𝑣 ∈ 𝔅ℝ ↦ (𝑢 × 𝑣)) |
22 | df-mpo 7163 | . . . . . . 7 ⊢ (𝑢 ∈ 𝔅ℝ, 𝑣 ∈ 𝔅ℝ ↦ (𝑢 × 𝑣)) = {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ) ∧ 𝑔 = (𝑢 × 𝑣))} | |
23 | 21, 22 | eqtri 2846 | . . . . . 6 ⊢ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ) ∧ 𝑔 = (𝑢 × 𝑣))} |
24 | 16, 18, 23 | 3sstr4i 4012 | . . . . 5 ⊢ 𝑅 ⊆ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) |
25 | rnss 5811 | . . . . 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 31407 | . . . 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 690 | . . 3 ⊢ ran 𝑅 ⊆ (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) |
29 | sigagenss2 31411 | . . 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 1457 | . 2 ⊢ (sigaGen‘ran 𝑅) ⊆ (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) |
31 | 1, 2, 3 | sxbrsigalem4 31547 | . 2 ⊢ (sigaGen‘(𝐽 ×t 𝐽)) = (sigaGen‘ran 𝑅) |
32 | eqid 2823 | . . . 4 ⊢ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) | |
33 | 32 | sxval 31451 | . . 3 ⊢ ((𝔅ℝ ∈ (sigAlgebra‘ℝ) ∧ 𝔅ℝ ∈ (sigAlgebra‘ℝ)) → (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)))) |
34 | 7, 7, 33 | mp2an 690 | . 2 ⊢ (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) |
35 | 30, 31, 34 | 3sstr4i 4012 | 1 ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅ℝ ×s 𝔅ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ∪ cuni 4840 × cxp 5555 ran crn 5558 ‘cfv 6357 (class class class)co 7158 {coprab 7159 ∈ cmpo 7160 ℝcr 10538 1c1 10540 + caddc 10542 / cdiv 11299 2c2 11695 ℤcz 11984 (,)cioo 12741 [,)cico 12743 ↑cexp 13432 topGenctg 16713 ×t ctx 22170 sigAlgebracsiga 31369 sigaGencsigagen 31399 𝔅ℝcbrsiga 31442 ×s csx 31449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-ac2 9887 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-ac 9544 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-refld 20751 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-fcls 22551 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-cfil 23860 df-cmet 23862 df-cms 23940 df-limc 24466 df-dv 24467 df-log 25142 df-cxp 25143 df-logb 25345 df-siga 31370 df-sigagen 31400 df-brsiga 31443 df-sx 31450 |
This theorem is referenced by: sxbrsigalem6 31549 |
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