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| Mirrors > Home > ILE Home > Th. List > opifismgmdc | GIF version | ||
| Description: A structure with a group addition operation expressed by a conditional operator is a magma if both values of the conditional operator are contained in the base set. (Contributed by AV, 9-Feb-2020.) |
| Ref | Expression |
|---|---|
| opifismgm.b | ⊢ 𝐵 = (Base‘𝑀) |
| opifismgm.p | ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝜓, 𝐶, 𝐷)) |
| opifismgmdc.dc | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → DECID 𝜓) |
| opifismgm.m | ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐵) |
| opifismgm.c | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵) |
| opifismgm.d | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| opifismgmdc | ⊢ (𝜑 → 𝑀 ∈ Mgm) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opifismgm.c | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵) | |
| 2 | opifismgm.d | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ 𝐵) | |
| 3 | opifismgmdc.dc | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → DECID 𝜓) | |
| 4 | 1, 2, 3 | ifcldcd 3608 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → if(𝜓, 𝐶, 𝐷) ∈ 𝐵) |
| 5 | 4 | ralrimivva 2588 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵) |
| 6 | 5 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵) |
| 7 | simprl 529 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵) | |
| 8 | simprr 531 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵) | |
| 9 | opifismgm.p | . . . . 5 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝜓, 𝐶, 𝐷)) | |
| 10 | 9 | ovmpoelrn 6293 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) |
| 11 | 6, 7, 8, 10 | syl3anc 1250 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) |
| 12 | 11 | ralrimivva 2588 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) |
| 13 | opifismgm.m | . . 3 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐵) | |
| 14 | opifismgm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
| 15 | eqid 2205 | . . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
| 16 | 14, 15 | ismgmn0 13190 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)) |
| 17 | 16 | exlimiv 1621 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)) |
| 18 | 13, 17 | syl 14 | . 2 ⊢ (𝜑 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)) |
| 19 | 12, 18 | mpbird 167 | 1 ⊢ (𝜑 → 𝑀 ∈ Mgm) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 836 = wceq 1373 ∃wex 1515 ∈ wcel 2176 ∀wral 2484 ifcif 3571 ‘cfv 5271 (class class class)co 5944 ∈ cmpo 5946 Basecbs 12832 +gcplusg 12909 Mgmcmgm 13186 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-cnex 8016 ax-resscn 8017 ax-1re 8019 ax-addrcl 8022 |
| This theorem depends on definitions: df-bi 117 df-dc 837 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-un 3170 df-in 3172 df-ss 3179 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-fv 5279 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-inn 9037 df-2 9095 df-ndx 12835 df-slot 12836 df-base 12838 df-plusg 12922 df-mgm 13188 |
| This theorem is referenced by: (None) |
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