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Mirrors > Home > HSE Home > Th. List > shscom | Structured version Visualization version GIF version |
Description: Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shscom | ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shel 28408 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℋ) | |
2 | shel 28408 | . . . . . . . . 9 ⊢ ((𝐵 ∈ Sℋ ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ℋ) | |
3 | 1, 2 | anim12i 600 | . . . . . . . 8 ⊢ (((𝐴 ∈ Sℋ ∧ 𝑦 ∈ 𝐴) ∧ (𝐵 ∈ Sℋ ∧ 𝑧 ∈ 𝐵)) → (𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ)) |
4 | 3 | an4s 639 | . . . . . . 7 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ)) |
5 | ax-hvcom 28198 | . . . . . . 7 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 +ℎ 𝑧) = (𝑧 +ℎ 𝑦)) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑦 +ℎ 𝑧) = (𝑧 +ℎ 𝑦)) |
7 | 6 | eqeq2d 2781 | . . . . 5 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑥 = (𝑦 +ℎ 𝑧) ↔ 𝑥 = (𝑧 +ℎ 𝑦))) |
8 | 7 | 2rexbidva 3204 | . . . 4 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑦))) |
9 | rexcom 3247 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑦) ↔ ∃𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑦)) | |
10 | 8, 9 | syl6bb 276 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧) ↔ ∃𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑦))) |
11 | shsel 28513 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝑥 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧))) | |
12 | shsel 28513 | . . . 4 ⊢ ((𝐵 ∈ Sℋ ∧ 𝐴 ∈ Sℋ ) → (𝑥 ∈ (𝐵 +ℋ 𝐴) ↔ ∃𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑦))) | |
13 | 12 | ancoms 455 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝑥 ∈ (𝐵 +ℋ 𝐴) ↔ ∃𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑦))) |
14 | 10, 11, 13 | 3bitr4d 300 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝑥 ∈ (𝐴 +ℋ 𝐵) ↔ 𝑥 ∈ (𝐵 +ℋ 𝐴))) |
15 | 14 | eqrdv 2769 | 1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∃wrex 3062 (class class class)co 6793 ℋchil 28116 +ℎ cva 28117 Sℋ csh 28125 +ℋ cph 28128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-hilex 28196 ax-hfvadd 28197 ax-hvcom 28198 ax-hvass 28199 ax-hv0cl 28200 ax-hvaddid 28201 ax-hfvmul 28202 ax-hvmulid 28203 ax-hvdistr2 28206 ax-hvmul0 28207 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-ltxr 10281 df-sub 10470 df-neg 10471 df-grpo 27687 df-ablo 27739 df-hvsub 28168 df-sh 28404 df-shs 28507 |
This theorem is referenced by: shsel2 28521 shsub2 28524 shscomi 28562 pjpjpre 28618 chscllem1 28836 chscllem2 28837 chscllem3 28838 chscllem4 28839 |
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