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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 2898 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝐶 ∈ 𝑢 ↔ 𝐶 ∈ 𝑈)) | |
3 | eleq2 2898 | . . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢 ↔ (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈)) | |
4 | 3 | imbi2d 342 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢) ↔ (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈))) |
5 | 4 | rexralbidv 3298 | . . . 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 346 | . . 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 22668 | . . . . 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 233 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ 𝐵 ∧ ∀𝑢 ∈ 𝐽 (𝐶 ∈ 𝑢 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢)))) |
17 | 16 | simprd 496 | . . 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 3622 | . 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 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 ↾ cres 5550 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 Basecbs 16471 TopOpenctopn 16683 Σg cgsu 16702 CMndccmn 18835 TopSpctps 21468 tsums ctsu 22661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-0g 16703 df-gsum 16704 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-cntz 18385 df-cmn 18837 df-fbas 20470 df-fg 20471 df-top 21430 df-topon 21447 df-topsp 21469 df-ntr 21556 df-nei 21634 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-tsms 22662 |
This theorem is referenced by: tsmsxplem1 22688 |
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