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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 3184 | . . 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 19271 | . . . 4 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵) |
7 | oveq2 7166 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 )) | |
8 | 7 | eqeq1d 2825 | . . . . 5 ⊢ (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍)) |
9 | 8 | rspcv 3620 | . . . 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 19278 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑍 ∈ 𝐵) → (𝑍 · 0 ) = 0 ) |
15 | 3, 12, 14 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑍 · 0 ) = 0 ) |
16 | 11, 15 | eqtr3d 2860 | 1 ⊢ (𝜑 → 𝑍 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 0gc0g 16715 SRingcsrg 19257 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-riota 7116 df-ov 7161 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-cmn 18910 df-srg 19258 |
This theorem is referenced by: (None) |
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