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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 28991 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℋ) | |
2 | shel 28991 | . . . . . . . . 9 ⊢ ((𝐵 ∈ Sℋ ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ℋ) | |
3 | 1, 2 | anim12i 614 | . . . . . . . 8 ⊢ (((𝐴 ∈ Sℋ ∧ 𝑦 ∈ 𝐴) ∧ (𝐵 ∈ Sℋ ∧ 𝑧 ∈ 𝐵)) → (𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ)) |
4 | 3 | an4s 658 | . . . . . . 7 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ)) |
5 | ax-hvcom 28781 | . . . . . . 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 2835 | . . . . 5 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑥 = (𝑦 +ℎ 𝑧) ↔ 𝑥 = (𝑧 +ℎ 𝑦))) |
8 | 7 | 2rexbidva 3302 | . . . 4 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑦))) |
9 | rexcom 3358 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑦) ↔ ∃𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑦)) | |
10 | 8, 9 | syl6bb 289 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧) ↔ ∃𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑦))) |
11 | shsel 29094 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝑥 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 +ℎ 𝑧))) | |
12 | shsel 29094 | . . . 4 ⊢ ((𝐵 ∈ Sℋ ∧ 𝐴 ∈ Sℋ ) → (𝑥 ∈ (𝐵 +ℋ 𝐴) ↔ ∃𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑦))) | |
13 | 12 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝑥 ∈ (𝐵 +ℋ 𝐴) ↔ ∃𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑦))) |
14 | 10, 11, 13 | 3bitr4d 313 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝑥 ∈ (𝐴 +ℋ 𝐵) ↔ 𝑥 ∈ (𝐵 +ℋ 𝐴))) |
15 | 14 | eqrdv 2822 | 1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 (class class class)co 7159 ℋchba 28699 +ℎ cva 28700 Sℋ csh 28708 +ℋ cph 28711 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-hilex 28779 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvdistr2 28789 ax-hvmul0 28790 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-sub 10875 df-neg 10876 df-grpo 28273 df-ablo 28325 df-hvsub 28751 df-sh 28987 df-shs 29088 |
This theorem is referenced by: shsel2 29102 shsub2 29105 shscomi 29143 pjpjpre 29199 chscllem1 29417 chscllem2 29418 chscllem3 29419 chscllem4 29420 |
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