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Mirrors > Home > HSE Home > Th. List > hvpncan | Structured version Visualization version GIF version |
Description: Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvpncan | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) −ℎ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvaddcl 28781 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) | |
2 | hvsubval 28785 | . . 3 ⊢ (((𝐴 +ℎ 𝐵) ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) −ℎ 𝐵) = ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵))) | |
3 | 1, 2 | sylancom 590 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) −ℎ 𝐵) = ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵))) |
4 | neg1cn 11743 | . . . . 5 ⊢ -1 ∈ ℂ | |
5 | hvmulcl 28782 | . . . . 5 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ 𝐵) ∈ ℋ) | |
6 | 4, 5 | mpan 688 | . . . 4 ⊢ (𝐵 ∈ ℋ → (-1 ·ℎ 𝐵) ∈ ℋ) |
7 | 6 | ancli 551 | . . 3 ⊢ (𝐵 ∈ ℋ → (𝐵 ∈ ℋ ∧ (-1 ·ℎ 𝐵) ∈ ℋ)) |
8 | ax-hvass 28771 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (-1 ·ℎ 𝐵) ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵)))) | |
9 | 8 | 3expb 1115 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ (-1 ·ℎ 𝐵) ∈ ℋ)) → ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵)))) |
10 | 7, 9 | sylan2 594 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐵)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵)))) |
11 | hvnegid 28796 | . . . 4 ⊢ (𝐵 ∈ ℋ → (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ) | |
12 | 11 | oveq2d 7164 | . . 3 ⊢ (𝐵 ∈ ℋ → (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ 0ℎ)) |
13 | ax-hvaddid 28773 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
14 | 12, 13 | sylan9eqr 2876 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = 𝐴) |
15 | 3, 10, 14 | 3eqtrd 2858 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐵) −ℎ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 (class class class)co 7148 ℂcc 10527 1c1 10530 -cneg 10863 ℋchba 28688 +ℎ cva 28689 ·ℎ csm 28690 0ℎc0v 28693 −ℎ cmv 28694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-hfvadd 28769 ax-hvass 28771 ax-hvaddid 28773 ax-hfvmul 28774 ax-hvmulid 28775 ax-hvdistr2 28778 ax-hvmul0 28779 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-ltxr 10672 df-sub 10864 df-neg 10865 df-hvsub 28740 |
This theorem is referenced by: hvpncan2 28809 mayete3i 29497 lnop0 29735 |
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