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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 29069 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵)) | |
2 | 1 | oveq1d 7206 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((𝐴 +ℎ 𝐵) +ℎ 𝐶) +ℎ 𝐷) = (((𝐴 +ℎ 𝐶) +ℎ 𝐵) +ℎ 𝐷)) |
3 | 2 | 3expa 1120 | . . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐶 ∈ ℋ) → (((𝐴 +ℎ 𝐵) +ℎ 𝐶) +ℎ 𝐷) = (((𝐴 +ℎ 𝐶) +ℎ 𝐵) +ℎ 𝐷)) |
4 | 3 | adantrr 717 | . 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 29047 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) | |
6 | ax-hvass 29037 | . . . 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 1122 | . . 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 583 | . 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 29047 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ 𝐶) ∈ ℋ) | |
10 | ax-hvass 29037 | . . . . 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 1122 | . . . 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 583 | . . 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 660 | . 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 2779 | 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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 (class class class)co 7191 ℋchba 28954 +ℎ cva 28955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-hfvadd 29035 ax-hvcom 29036 ax-hvass 29037 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 |
This theorem is referenced by: hvsub4 29072 hvadd4i 29093 shscli 29352 spanunsni 29614 mayete3i 29763 lnophsi 30036 |
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