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Mirrors > Home > HSE Home > Th. List > shintcl | Structured version Visualization version GIF version |
Description: The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shintcl | ⊢ ((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Sℋ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inteq 4785 | . . 3 ⊢ (𝐴 = if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) → ∩ 𝐴 = ∩ if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ )) | |
2 | 1 | eleq1d 2867 | . 2 ⊢ (𝐴 = if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) → (∩ 𝐴 ∈ Sℋ ↔ ∩ if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ∈ Sℋ )) |
3 | sseq1 3913 | . . . . 5 ⊢ (𝐴 = if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) → (𝐴 ⊆ Sℋ ↔ if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ⊆ Sℋ )) | |
4 | neeq1 3046 | . . . . 5 ⊢ (𝐴 = if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) → (𝐴 ≠ ∅ ↔ if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ≠ ∅)) | |
5 | 3, 4 | anbi12d 630 | . . . 4 ⊢ (𝐴 = if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) → ((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) ↔ (if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ⊆ Sℋ ∧ if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ≠ ∅))) |
6 | sseq1 3913 | . . . . 5 ⊢ ( Sℋ = if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) → ( Sℋ ⊆ Sℋ ↔ if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ⊆ Sℋ )) | |
7 | neeq1 3046 | . . . . 5 ⊢ ( Sℋ = if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) → ( Sℋ ≠ ∅ ↔ if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ≠ ∅)) | |
8 | 6, 7 | anbi12d 630 | . . . 4 ⊢ ( Sℋ = if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) → (( Sℋ ⊆ Sℋ ∧ Sℋ ≠ ∅) ↔ (if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ⊆ Sℋ ∧ if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ≠ ∅))) |
9 | ssid 3910 | . . . . 5 ⊢ Sℋ ⊆ Sℋ | |
10 | h0elsh 28724 | . . . . . 6 ⊢ 0ℋ ∈ Sℋ | |
11 | 10 | ne0ii 4223 | . . . . 5 ⊢ Sℋ ≠ ∅ |
12 | 9, 11 | pm3.2i 471 | . . . 4 ⊢ ( Sℋ ⊆ Sℋ ∧ Sℋ ≠ ∅) |
13 | 5, 8, 12 | elimhyp 4444 | . . 3 ⊢ (if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ⊆ Sℋ ∧ if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ≠ ∅) |
14 | 13 | shintcli 28797 | . 2 ⊢ ∩ if((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅), 𝐴, Sℋ ) ∈ Sℋ |
15 | 2, 14 | dedth 4437 | 1 ⊢ ((𝐴 ⊆ Sℋ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Sℋ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ⊆ wss 3859 ∅c0 4211 ifcif 4381 ∩ cint 4782 Sℋ csh 28396 0ℋc0h 28403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 ax-hilex 28467 ax-hfvadd 28468 ax-hvcom 28469 ax-hvass 28470 ax-hv0cl 28471 ax-hvaddid 28472 ax-hfvmul 28473 ax-hvmulid 28474 ax-hvmulass 28475 ax-hvdistr1 28476 ax-hvdistr2 28477 ax-hvmul0 28478 ax-hfi 28547 ax-his1 28550 ax-his2 28551 ax-his3 28552 ax-his4 28553 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-map 8258 df-pm 8259 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-n0 11746 df-z 11830 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-icc 12595 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-topgen 16546 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-top 21186 df-topon 21203 df-bases 21238 df-lm 21521 df-haus 21607 df-grpo 27961 df-gid 27962 df-ginv 27963 df-gdiv 27964 df-ablo 28013 df-vc 28027 df-nv 28060 df-va 28063 df-ba 28064 df-sm 28065 df-0v 28066 df-vs 28067 df-nmcv 28068 df-ims 28069 df-hnorm 28436 df-hvsub 28439 df-hlim 28440 df-sh 28675 df-ch 28689 df-ch0 28721 |
This theorem is referenced by: spancl 28804 shsval2i 28855 |
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