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Mirrors > Home > MPE Home > Th. List > Mathboxes > sxbrsigalem5 | Structured version Visualization version GIF version |
Description: First direction for sxbrsiga 29679. (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 29673 | . . . 4 ⊢ ∪ ran 𝑅 = (ℝ × ℝ) |
5 | br2base 29658 | . . . 4 ⊢ ∪ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = (ℝ × ℝ) | |
6 | 4, 5 | eqtr4i 2635 | . . 3 ⊢ ∪ ran 𝑅 = ∪ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) |
7 | brsigarn 29574 | . . . . . . 7 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
8 | 7 | elexi 3186 | . . . . . 6 ⊢ 𝔅ℝ ∈ V |
9 | 8, 8 | mpt2ex 7136 | . . . . 5 ⊢ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) ∈ V |
10 | 9 | rnex 6992 | . . . 4 ⊢ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) ∈ V |
11 | 1, 2 | dya2icobrsiga 29665 | . . . . . . . . . 10 ⊢ ran 𝐼 ⊆ 𝔅ℝ |
12 | 11 | sseli 3564 | . . . . . . . . 9 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ∈ 𝔅ℝ) |
13 | 11 | sseli 3564 | . . . . . . . . 9 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ∈ 𝔅ℝ) |
14 | 12, 13 | anim12i 588 | . . . . . . . 8 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ)) |
15 | 14 | anim1i 590 | . . . . . . 7 ⊢ (((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑔 = (𝑢 × 𝑣)) → ((𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ) ∧ 𝑔 = (𝑢 × 𝑣))) |
16 | 15 | ssoprab2i 6647 | . . . . . 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-mpt2 6554 | . . . . . . 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 2632 | . . . . . 6 ⊢ 𝑅 = {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑔 = (𝑢 × 𝑣))} |
19 | xpeq1 5052 | . . . . . . . 8 ⊢ (𝑒 = 𝑢 → (𝑒 × 𝑓) = (𝑢 × 𝑓)) | |
20 | xpeq2 5053 | . . . . . . . 8 ⊢ (𝑓 = 𝑣 → (𝑢 × 𝑓) = (𝑢 × 𝑣)) | |
21 | 19, 20 | cbvmpt2v 6633 | . . . . . . 7 ⊢ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = (𝑢 ∈ 𝔅ℝ, 𝑣 ∈ 𝔅ℝ ↦ (𝑢 × 𝑣)) |
22 | df-mpt2 6554 | . . . . . . 7 ⊢ (𝑢 ∈ 𝔅ℝ, 𝑣 ∈ 𝔅ℝ ↦ (𝑢 × 𝑣)) = {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ) ∧ 𝑔 = (𝑢 × 𝑣))} | |
23 | 21, 22 | eqtri 2632 | . . . . . 6 ⊢ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = {〈〈𝑢, 𝑣〉, 𝑔〉 ∣ ((𝑢 ∈ 𝔅ℝ ∧ 𝑣 ∈ 𝔅ℝ) ∧ 𝑔 = (𝑢 × 𝑣))} |
24 | 16, 18, 23 | 3sstr4i 3607 | . . . . 5 ⊢ 𝑅 ⊆ (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) |
25 | rnss 5275 | . . . . 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 29536 | . . . 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 704 | . . 3 ⊢ ran 𝑅 ⊆ (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) |
29 | sigagenss2 29540 | . . 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 1416 | . 2 ⊢ (sigaGen‘ran 𝑅) ⊆ (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) |
31 | 1, 2, 3 | sxbrsigalem4 29676 | . 2 ⊢ (sigaGen‘(𝐽 ×t 𝐽)) = (sigaGen‘ran 𝑅) |
32 | eqid 2610 | . . . 4 ⊢ ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) = ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)) | |
33 | 32 | sxval 29580 | . . 3 ⊢ ((𝔅ℝ ∈ (sigAlgebra‘ℝ) ∧ 𝔅ℝ ∈ (sigAlgebra‘ℝ)) → (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓)))) |
34 | 7, 7, 33 | mp2an 704 | . 2 ⊢ (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘ran (𝑒 ∈ 𝔅ℝ, 𝑓 ∈ 𝔅ℝ ↦ (𝑒 × 𝑓))) |
35 | 30, 31, 34 | 3sstr4i 3607 | 1 ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅ℝ ×s 𝔅ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∪ cuni 4372 × cxp 5036 ran crn 5039 ‘cfv 5804 (class class class)co 6549 {coprab 6550 ↦ cmpt2 6551 ℝcr 9814 1c1 9816 + caddc 9818 / cdiv 10563 2c2 10947 ℤcz 11254 (,)cioo 12046 [,)cico 12048 ↑cexp 12722 topGenctg 15921 ×t ctx 21173 sigAlgebracsiga 29497 sigaGencsigagen 29528 𝔅ℝcbrsiga 29571 ×s csx 29578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-refld 19770 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-fcls 21555 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-cfil 22861 df-cmet 22863 df-cms 22940 df-limc 23436 df-dv 23437 df-log 24107 df-cxp 24108 df-logb 24303 df-siga 29498 df-sigagen 29529 df-brsiga 29572 df-sx 29579 |
This theorem is referenced by: sxbrsigalem6 29678 |
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