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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 2826 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝐶 ∈ 𝑢 ↔ 𝐶 ∈ 𝑈)) | |
3 | eleq2 2826 | . . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢 ↔ (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈)) | |
4 | 3 | imbi2d 340 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢) ↔ (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈))) |
5 | 4 | rexralbidv 3212 | . . . 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 344 | . . 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 23468 | . . . . 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 231 | . . . 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 3580 | . 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 1541 ∈ wcel 2106 ∀wral 3062 ∃wrex 3071 ∩ cin 3907 ⊆ wss 3908 𝒫 cpw 4558 ↾ cres 5633 ⟶wf 6489 ‘cfv 6493 (class class class)co 7353 Fincfn 8879 Basecbs 17075 TopOpenctopn 17295 Σg cgsu 17314 CMndccmn 19553 TopSpctps 22265 tsums ctsu 23461 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-fzo 13560 df-seq 13899 df-hash 14223 df-0g 17315 df-gsum 17316 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-cntz 19088 df-cmn 19555 df-fbas 20778 df-fg 20779 df-top 22227 df-topon 22244 df-topsp 22266 df-ntr 22355 df-nei 22433 df-fil 23181 df-fm 23273 df-flim 23274 df-flf 23275 df-tsms 23462 |
This theorem is referenced by: tsmsxplem1 23488 |
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