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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgmplusfreseq | Structured version Visualization version GIF version |
Description: If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
Ref | Expression |
---|---|
plusfreseq.1 | ⊢ 𝐵 = (Base‘𝑀) |
plusfreseq.2 | ⊢ + = (+g‘𝑀) |
plusfreseq.3 | ⊢ ⨣ = (+𝑓‘𝑀) |
Ref | Expression |
---|---|
mgmplusfreseq | ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plusfreseq.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
2 | plusfreseq.3 | . . . . 5 ⊢ ⨣ = (+𝑓‘𝑀) | |
3 | 1, 2 | mgmplusf 18461 | . . . 4 ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵) |
4 | frn 6672 | . . . 4 ⊢ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 → ran ⨣ ⊆ 𝐵) | |
5 | ssel 3935 | . . . . 5 ⊢ (ran ⨣ ⊆ 𝐵 → (∅ ∈ ran ⨣ → ∅ ∈ 𝐵)) | |
6 | 5 | nelcon3d 3059 | . . . 4 ⊢ (ran ⨣ ⊆ 𝐵 → (∅ ∉ 𝐵 → ∅ ∉ ran ⨣ )) |
7 | 3, 4, 6 | 3syl 18 | . . 3 ⊢ (𝑀 ∈ Mgm → (∅ ∉ 𝐵 → ∅ ∉ ran ⨣ )) |
8 | 7 | imp 407 | . 2 ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ∅ ∉ ran ⨣ ) |
9 | plusfreseq.2 | . . 3 ⊢ + = (+g‘𝑀) | |
10 | 1, 9, 2 | plusfreseq 45961 | . 2 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
11 | 8, 10 | syl 17 | 1 ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∉ wnel 3047 ⊆ wss 3908 ∅c0 4280 × cxp 5629 ran crn 5632 ↾ cres 5633 ⟶wf 6489 ‘cfv 6493 Basecbs 17037 +gcplusg 17087 +𝑓cplusf 18448 Mgmcmgm 18449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-plusf 18450 df-mgm 18451 |
This theorem is referenced by: (None) |
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