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Mirrors > Home > HSE Home > Th. List > hvadd4 | Structured version Visualization version GIF version |
Description: Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvadd4 | ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷)) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvadd32 30782 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵)) | |
2 | 1 | oveq1d 7417 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((𝐴 +ℎ 𝐵) +ℎ 𝐶) +ℎ 𝐷) = (((𝐴 +ℎ 𝐶) +ℎ 𝐵) +ℎ 𝐷)) |
3 | 2 | 3expa 1115 | . . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐶 ∈ ℋ) → (((𝐴 +ℎ 𝐵) +ℎ 𝐶) +ℎ 𝐷) = (((𝐴 +ℎ 𝐶) +ℎ 𝐵) +ℎ 𝐷)) |
4 | 3 | adantrr 714 | . 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 +ℎ 𝐵) +ℎ 𝐶) +ℎ 𝐷) = (((𝐴 +ℎ 𝐶) +ℎ 𝐵) +ℎ 𝐷)) |
5 | hvaddcl 30760 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) | |
6 | ax-hvass 30750 | . . . 4 ⊢ (((𝐴 +ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((𝐴 +ℎ 𝐵) +ℎ 𝐶) +ℎ 𝐷) = ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷))) | |
7 | 6 | 3expb 1117 | . . 3 ⊢ (((𝐴 +ℎ 𝐵) ∈ ℋ ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 +ℎ 𝐵) +ℎ 𝐶) +ℎ 𝐷) = ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷))) |
8 | 5, 7 | sylan 579 | . 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 +ℎ 𝐵) +ℎ 𝐶) +ℎ 𝐷) = ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷))) |
9 | hvaddcl 30760 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ 𝐶) ∈ ℋ) | |
10 | ax-hvass 30750 | . . . . 5 ⊢ (((𝐴 +ℎ 𝐶) ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((𝐴 +ℎ 𝐶) +ℎ 𝐵) +ℎ 𝐷) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷))) | |
11 | 10 | 3expb 1117 | . . . 4 ⊢ (((𝐴 +ℎ 𝐶) ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 +ℎ 𝐶) +ℎ 𝐵) +ℎ 𝐷) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷))) |
12 | 9, 11 | sylan 579 | . . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ (𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 +ℎ 𝐶) +ℎ 𝐵) +ℎ 𝐷) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷))) |
13 | 12 | an4s 657 | . 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 +ℎ 𝐶) +ℎ 𝐵) +ℎ 𝐷) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷))) |
14 | 4, 8, 13 | 3eqtr3d 2772 | 1 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷)) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 (class class class)co 7402 ℋchba 30667 +ℎ cva 30668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-hfvadd 30748 ax-hvcom 30749 ax-hvass 30750 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 |
This theorem is referenced by: hvsub4 30785 hvadd4i 30806 shscli 31065 spanunsni 31327 mayete3i 31476 lnophsi 31749 |
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