![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > srgisid | Structured version Visualization version GIF version |
Description: In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.) |
Ref | Expression |
---|---|
srgz.b | ⊢ 𝐵 = (Base‘𝑅) |
srgz.t | ⊢ · = (.r‘𝑅) |
srgz.z | ⊢ 0 = (0g‘𝑅) |
srgisid.1 | ⊢ (𝜑 → 𝑅 ∈ SRing) |
srgisid.2 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
srgisid.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍 · 𝑥) = 𝑍) |
Ref | Expression |
---|---|
srgisid | ⊢ (𝜑 → 𝑍 = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srgisid.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍 · 𝑥) = 𝑍) | |
2 | 1 | ralrimiva 3136 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍) |
3 | srgisid.1 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ SRing) | |
4 | srgz.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
5 | srgz.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
6 | 4, 5 | srg0cl 20179 | . . . 4 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵) |
7 | oveq2 7424 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 )) | |
8 | 7 | eqeq1d 2728 | . . . . 5 ⊢ (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍)) |
9 | 8 | rspcv 3603 | . . . 4 ⊢ ( 0 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍)) |
10 | 3, 6, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍)) |
11 | 2, 10 | mpd 15 | . 2 ⊢ (𝜑 → (𝑍 · 0 ) = 𝑍) |
12 | srgisid.2 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
13 | srgz.t | . . . 4 ⊢ · = (.r‘𝑅) | |
14 | 4, 13, 5 | srgrz 20186 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑍 ∈ 𝐵) → (𝑍 · 0 ) = 0 ) |
15 | 3, 12, 14 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝑍 · 0 ) = 0 ) |
16 | 11, 15 | eqtr3d 2768 | 1 ⊢ (𝜑 → 𝑍 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 .rcmulr 17262 0gc0g 17449 SRingcsrg 20165 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fv 6554 df-riota 7372 df-ov 7419 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-cmn 19776 df-srg 20166 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |