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Mirrors > Home > HSE Home > Th. List > hvaddcan | Structured version Visualization version GIF version |
Description: Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvaddcan | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7433 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 +ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵)) | |
2 | oveq1 7433 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 +ℎ 𝐶) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶)) | |
3 | 1, 2 | eqeq12d 2744 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶))) |
4 | 3 | bibi1d 342 | . 2 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) ↔ 𝐵 = 𝐶) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶) ↔ 𝐵 = 𝐶))) |
5 | oveq2 7434 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) | |
6 | 5 | eqeq1d 2730 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶))) |
7 | eqeq1 2732 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (𝐵 = 𝐶 ↔ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) = 𝐶)) | |
8 | 6, 7 | bibi12d 344 | . 2 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶) ↔ 𝐵 = 𝐶) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶) ↔ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) = 𝐶))) |
9 | oveq2 7434 | . . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) | |
10 | 9 | eqeq2d 2739 | . . 3 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶) ↔ (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ)))) |
11 | eqeq2 2740 | . . 3 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) = 𝐶 ↔ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) = if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) | |
12 | 10, 11 | bibi12d 344 | . 2 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶) ↔ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) = 𝐶) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ)) ↔ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) = if(𝐶 ∈ ℋ, 𝐶, 0ℎ)))) |
13 | ifhvhv0 30860 | . . 3 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ | |
14 | ifhvhv0 30860 | . . 3 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ∈ ℋ | |
15 | ifhvhv0 30860 | . . 3 ⊢ if(𝐶 ∈ ℋ, 𝐶, 0ℎ) ∈ ℋ | |
16 | 13, 14, 15 | hvaddcani 30903 | . 2 ⊢ ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ)) ↔ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) = if(𝐶 ∈ ℋ, 𝐶, 0ℎ)) |
17 | 4, 8, 12, 16 | dedth3h 4592 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) ↔ 𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ifcif 4532 (class class class)co 7426 ℋchba 30757 +ℎ cva 30758 0ℎc0v 30762 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-hvcom 30839 ax-hvass 30840 ax-hv0cl 30841 ax-hvaddid 30842 ax-hfvmul 30843 ax-hvmulid 30844 ax-hvdistr2 30847 ax-hvmul0 30848 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-ltxr 11293 df-sub 11486 df-neg 11487 df-hvsub 30809 |
This theorem is referenced by: hvaddcan2 30909 hvsubcan 30912 |
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