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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 2828 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝐶 ∈ 𝑢 ↔ 𝐶 ∈ 𝑈)) | |
3 | eleq2 2828 | . . . . . 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 3221 | . . . 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 24157 | . . . . 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 232 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ 𝐵 ∧ ∀𝑢 ∈ 𝐽 (𝐶 ∈ 𝑢 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢)))) |
17 | 16 | simprd 495 | . . 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 3623 | . 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 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 Basecbs 17245 TopOpenctopn 17468 Σg cgsu 17487 CMndccmn 19813 TopSpctps 22954 tsums ctsu 24150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-cntz 19348 df-cmn 19815 df-fbas 21379 df-fg 21380 df-top 22916 df-topon 22933 df-topsp 22955 df-ntr 23044 df-nei 23122 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tsms 24151 |
This theorem is referenced by: tsmsxplem1 24177 |
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