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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 7416 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 +ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵)) | |
2 | oveq1 7416 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 +ℎ 𝐶) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐶)) | |
3 | 1, 2 | eqeq12d 2749 | . . 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 344 | . 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 7417 | . . . 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 2735 | . . 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 2737 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (𝐵 = 𝐶 ↔ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) = 𝐶)) | |
8 | 6, 7 | bibi12d 346 | . 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 7417 | . . . 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 2744 | . . 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 2745 | . . 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 346 | . 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 30275 | . . 3 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ | |
14 | ifhvhv0 30275 | . . 3 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ∈ ℋ | |
15 | ifhvhv0 30275 | . . 3 ⊢ if(𝐶 ∈ ℋ, 𝐶, 0ℎ) ∈ ℋ | |
16 | 13, 14, 15 | hvaddcani 30318 | . 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 4589 | 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 1088 = wceq 1542 ∈ wcel 2107 ifcif 4529 (class class class)co 7409 ℋchba 30172 +ℎ cva 30173 0ℎc0v 30177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-hvcom 30254 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvdistr2 30262 ax-hvmul0 30263 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-sub 11446 df-neg 11447 df-hvsub 30224 |
This theorem is referenced by: hvaddcan2 30324 hvsubcan 30327 |
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