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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 10638 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1 | tpid2 4709 | . . . . 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 31826 | . . . . 5 ⊢ ((0 ∈ {-1, 0, 1} ∧ 𝑢 ∈ {-1, 0, 1}) → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢)) |
5 | 2, 4 | mpan 688 | . . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢)) |
6 | iftrue 4476 | . . . . . 6 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0) | |
7 | id 22 | . . . . . 6 ⊢ (𝑢 = 0 → 𝑢 = 0) | |
8 | 6, 7 | eqtr4d 2862 | . . . . 5 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢) |
9 | iffalse 4479 | . . . . 5 ⊢ (¬ 𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢) | |
10 | 8, 9 | pm2.61i 184 | . . . 4 ⊢ if(𝑢 = 0, 0, 𝑢) = 𝑢 |
11 | 5, 10 | syl6eq 2875 | . . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = 𝑢) |
12 | 3 | signspval 31826 | . . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0)) |
13 | 2, 12 | mpan2 689 | . . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0)) |
14 | eqid 2824 | . . . . 5 ⊢ 0 = 0 | |
15 | 14 | iftruei 4477 | . . . 4 ⊢ if(0 = 0, 𝑢, 0) = 𝑢 |
16 | 13, 15 | syl6eq 2875 | . . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = 𝑢) |
17 | 11, 16 | jca 514 | . 2 ⊢ (𝑢 ∈ {-1, 0, 1} → ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)) |
18 | 17 | rgen 3151 | 1 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ifcif 4470 {ctp 4574 (class class class)co 7159 ∈ cmpo 7161 0cc0 10540 1c1 10541 -cneg 10874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-mulcl 10602 ax-i2m1 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 |
This theorem is referenced by: signsw0g 31830 signswmnd 31831 |
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