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Mirrors > Home > MPE Home > Th. List > Mathboxes > signsw0glem | Structured version Visualization version GIF version |
Description: Neutral element property of ⨣. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
Ref | Expression |
---|---|
signsw.p | ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) |
Ref | Expression |
---|---|
signsw0glem | ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 11213 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1 | tpid2 4775 | . . . . 5 ⊢ 0 ∈ {-1, 0, 1} |
3 | signsw.p | . . . . . 6 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) | |
4 | 3 | signspval 33858 | . . . . 5 ⊢ ((0 ∈ {-1, 0, 1} ∧ 𝑢 ∈ {-1, 0, 1}) → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢)) |
5 | 2, 4 | mpan 687 | . . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢)) |
6 | iftrue 4535 | . . . . . 6 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0) | |
7 | id 22 | . . . . . 6 ⊢ (𝑢 = 0 → 𝑢 = 0) | |
8 | 6, 7 | eqtr4d 2774 | . . . . 5 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢) |
9 | iffalse 4538 | . . . . 5 ⊢ (¬ 𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢) | |
10 | 8, 9 | pm2.61i 182 | . . . 4 ⊢ if(𝑢 = 0, 0, 𝑢) = 𝑢 |
11 | 5, 10 | eqtrdi 2787 | . . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = 𝑢) |
12 | 3 | signspval 33858 | . . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0)) |
13 | 2, 12 | mpan2 688 | . . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0)) |
14 | eqid 2731 | . . . . 5 ⊢ 0 = 0 | |
15 | 14 | iftruei 4536 | . . . 4 ⊢ if(0 = 0, 𝑢, 0) = 𝑢 |
16 | 13, 15 | eqtrdi 2787 | . . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = 𝑢) |
17 | 11, 16 | jca 511 | . 2 ⊢ (𝑢 ∈ {-1, 0, 1} → ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)) |
18 | 17 | rgen 3062 | 1 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ifcif 4529 {ctp 4633 (class class class)co 7412 ∈ cmpo 7414 0cc0 11113 1c1 11114 -cneg 11450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-mulcl 11175 ax-i2m1 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7415 df-oprab 7416 df-mpo 7417 |
This theorem is referenced by: signsw0g 33862 signswmnd 33863 |
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