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Mirrors > Home > HSE Home > Th. List > spanun | Structured version Visualization version GIF version |
Description: The span of a union is the subspace sum of spans. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spanun | ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (span‘(𝐴 ∪ 𝐵)) = ((span‘𝐴) +ℋ (span‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 3722 | . . . 4 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (𝐴 ∪ 𝐵) = (if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵)) | |
2 | 1 | fveq2d 6107 | . . 3 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (span‘(𝐴 ∪ 𝐵)) = (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵))) |
3 | fveq2 6103 | . . . 4 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (span‘𝐴) = (span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ))) | |
4 | 3 | oveq1d 6564 | . . 3 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → ((span‘𝐴) +ℋ (span‘𝐵)) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘𝐵))) |
5 | 2, 4 | eqeq12d 2625 | . 2 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → ((span‘(𝐴 ∪ 𝐵)) = ((span‘𝐴) +ℋ (span‘𝐵)) ↔ (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵)) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘𝐵)))) |
6 | uneq2 3723 | . . . 4 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → (if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵) = (if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ if(𝐵 ⊆ ℋ, 𝐵, ℋ))) | |
7 | 6 | fveq2d 6107 | . . 3 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵)) = (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ if(𝐵 ⊆ ℋ, 𝐵, ℋ)))) |
8 | fveq2 6103 | . . . 4 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → (span‘𝐵) = (span‘if(𝐵 ⊆ ℋ, 𝐵, ℋ))) | |
9 | 8 | oveq2d 6565 | . . 3 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘𝐵)) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘if(𝐵 ⊆ ℋ, 𝐵, ℋ)))) |
10 | 7, 9 | eqeq12d 2625 | . 2 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → ((span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ 𝐵)) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘𝐵)) ↔ (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ if(𝐵 ⊆ ℋ, 𝐵, ℋ))) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘if(𝐵 ⊆ ℋ, 𝐵, ℋ))))) |
11 | sseq1 3589 | . . . 4 ⊢ (𝐴 = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → (𝐴 ⊆ ℋ ↔ if(𝐴 ⊆ ℋ, 𝐴, ℋ) ⊆ ℋ)) | |
12 | sseq1 3589 | . . . 4 ⊢ ( ℋ = if(𝐴 ⊆ ℋ, 𝐴, ℋ) → ( ℋ ⊆ ℋ ↔ if(𝐴 ⊆ ℋ, 𝐴, ℋ) ⊆ ℋ)) | |
13 | ssid 3587 | . . . 4 ⊢ ℋ ⊆ ℋ | |
14 | 11, 12, 13 | elimhyp 4096 | . . 3 ⊢ if(𝐴 ⊆ ℋ, 𝐴, ℋ) ⊆ ℋ |
15 | sseq1 3589 | . . . 4 ⊢ (𝐵 = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → (𝐵 ⊆ ℋ ↔ if(𝐵 ⊆ ℋ, 𝐵, ℋ) ⊆ ℋ)) | |
16 | sseq1 3589 | . . . 4 ⊢ ( ℋ = if(𝐵 ⊆ ℋ, 𝐵, ℋ) → ( ℋ ⊆ ℋ ↔ if(𝐵 ⊆ ℋ, 𝐵, ℋ) ⊆ ℋ)) | |
17 | 15, 16, 13 | elimhyp 4096 | . . 3 ⊢ if(𝐵 ⊆ ℋ, 𝐵, ℋ) ⊆ ℋ |
18 | 14, 17 | spanuni 27787 | . 2 ⊢ (span‘(if(𝐴 ⊆ ℋ, 𝐴, ℋ) ∪ if(𝐵 ⊆ ℋ, 𝐵, ℋ))) = ((span‘if(𝐴 ⊆ ℋ, 𝐴, ℋ)) +ℋ (span‘if(𝐵 ⊆ ℋ, 𝐵, ℋ))) |
19 | 5, 10, 18 | dedth2h 4090 | 1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (span‘(𝐴 ∪ 𝐵)) = ((span‘𝐴) +ℋ (span‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∪ cun 3538 ⊆ wss 3540 ifcif 4036 ‘cfv 5804 (class class class)co 6549 ℋchil 27160 +ℋ cph 27172 spancspn 27173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-lm 20843 df-haus 20929 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-hnorm 27209 df-hvsub 27212 df-hlim 27213 df-sh 27448 df-ch 27462 df-ch0 27494 df-shs 27551 df-span 27552 |
This theorem is referenced by: spanpr 27823 superpos 28597 |
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