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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 17864 | . . . 4 ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵) |
4 | frn 6522 | . . . 4 ⊢ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 → ran ⨣ ⊆ 𝐵) | |
5 | ssel 3963 | . . . . 5 ⊢ (ran ⨣ ⊆ 𝐵 → (∅ ∈ ran ⨣ → ∅ ∈ 𝐵)) | |
6 | 5 | nelcon3d 3137 | . . . 4 ⊢ (ran ⨣ ⊆ 𝐵 → (∅ ∉ 𝐵 → ∅ ∉ ran ⨣ )) |
7 | 3, 4, 6 | 3syl 18 | . . 3 ⊢ (𝑀 ∈ Mgm → (∅ ∉ 𝐵 → ∅ ∉ ran ⨣ )) |
8 | 7 | imp 409 | . 2 ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ∅ ∉ ran ⨣ ) |
9 | plusfreseq.2 | . . 3 ⊢ + = (+g‘𝑀) | |
10 | 1, 9, 2 | plusfreseq 44046 | . 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 398 = wceq 1537 ∈ wcel 2114 ∉ wnel 3125 ⊆ wss 3938 ∅c0 4293 × cxp 5555 ran crn 5558 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 Basecbs 16485 +gcplusg 16567 +𝑓cplusf 17851 Mgmcmgm 17852 |
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 ax-un 7463 |
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-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 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-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-plusf 17853 df-mgm 17854 |
This theorem is referenced by: (None) |
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