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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 2830 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝐶 ∈ 𝑢 ↔ 𝐶 ∈ 𝑈)) | |
| 3 | eleq2 2830 | . . . . . 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 3223 | . . . 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 24141 | . . . . 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 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 Basecbs 17247 TopOpenctopn 17466 Σg cgsu 17485 CMndccmn 19798 TopSpctps 22938 tsums ctsu 24134 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cntz 19335 df-cmn 19800 df-fbas 21361 df-fg 21362 df-top 22900 df-topon 22917 df-topsp 22939 df-ntr 23028 df-nei 23106 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tsms 24135 |
| This theorem is referenced by: tsmsxplem1 24161 |
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