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Mirrors > Home > MPE Home > Th. List > tsmsi | Structured version Visualization version GIF version |
Description: The property of being a sum of the sequence 𝐹 in the topological commutative monoid 𝐺. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
eltsms.b | ⊢ 𝐵 = (Base‘𝐺) |
eltsms.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
eltsms.s | ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) |
eltsms.1 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
eltsms.2 | ⊢ (𝜑 → 𝐺 ∈ TopSp) |
eltsms.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
eltsms.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
tsmsi.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐺 tsums 𝐹)) |
tsmsi.4 | ⊢ (𝜑 → 𝑈 ∈ 𝐽) |
tsmsi.5 | ⊢ (𝜑 → 𝐶 ∈ 𝑈) |
Ref | Expression |
---|---|
tsmsi | ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsmsi.5 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑈) | |
2 | eleq2 2819 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝐶 ∈ 𝑢 ↔ 𝐶 ∈ 𝑈)) | |
3 | eleq2 2819 | . . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢 ↔ (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈)) | |
4 | 3 | imbi2d 344 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢) ↔ (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈))) |
5 | 4 | rexralbidv 3210 | . . . 4 ⊢ (𝑢 = 𝑈 → (∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢) ↔ ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈))) |
6 | 2, 5 | imbi12d 348 | . . 3 ⊢ (𝑢 = 𝑈 → ((𝐶 ∈ 𝑢 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢)) ↔ (𝐶 ∈ 𝑈 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈)))) |
7 | tsmsi.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (𝐺 tsums 𝐹)) | |
8 | eltsms.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
9 | eltsms.j | . . . . . 6 ⊢ 𝐽 = (TopOpen‘𝐺) | |
10 | eltsms.s | . . . . . 6 ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) | |
11 | eltsms.1 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
12 | eltsms.2 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TopSp) | |
13 | eltsms.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
14 | eltsms.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
15 | 8, 9, 10, 11, 12, 13, 14 | eltsms 22984 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ (𝐶 ∈ 𝐵 ∧ ∀𝑢 ∈ 𝐽 (𝐶 ∈ 𝑢 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢))))) |
16 | 7, 15 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ 𝐵 ∧ ∀𝑢 ∈ 𝐽 (𝐶 ∈ 𝑢 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢)))) |
17 | 16 | simprd 499 | . . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐽 (𝐶 ∈ 𝑢 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢))) |
18 | tsmsi.4 | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐽) | |
19 | 6, 17, 18 | rspcdva 3529 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝑈 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈))) |
20 | 1, 19 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ∩ cin 3852 ⊆ wss 3853 𝒫 cpw 4499 ↾ cres 5538 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 Basecbs 16666 TopOpenctopn 16880 Σg cgsu 16899 CMndccmn 19124 TopSpctps 21783 tsums ctsu 22977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-0g 16900 df-gsum 16901 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-cntz 18665 df-cmn 19126 df-fbas 20314 df-fg 20315 df-top 21745 df-topon 21762 df-topsp 21784 df-ntr 21871 df-nei 21949 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-tsms 22978 |
This theorem is referenced by: tsmsxplem1 23004 |
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