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Mirrors > Home > MPE Home > Th. List > Mathboxes > plusfreseq | Structured version Visualization version GIF version |
Description: If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily 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 |
---|---|
plusfreseq | ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plusfreseq.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
2 | plusfreseq.3 | . . . . 5 ⊢ ⨣ = (+𝑓‘𝑀) | |
3 | 1, 2 | plusffn 17073 | . . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵) |
4 | fnfun 5902 | . . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → Fun ⨣ ) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ Fun ⨣ |
6 | 5 | a1i 11 | . 2 ⊢ (∅ ∉ ran ⨣ → Fun ⨣ ) |
7 | id 22 | . 2 ⊢ (∅ ∉ ran ⨣ → ∅ ∉ ran ⨣ ) | |
8 | plusfreseq.2 | . . . . . . 7 ⊢ + = (+g‘𝑀) | |
9 | 1, 8, 2 | plusfval 17071 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦)) |
10 | 9 | eqcomd 2616 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) |
11 | 10 | rgen2a 2960 | . . . 4 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦) |
12 | 11 | a1i 11 | . . 3 ⊢ (∅ ∉ ran ⨣ → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) |
13 | fveq2 6103 | . . . . . 6 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( + ‘𝑝) = ( + ‘〈𝑥, 𝑦〉)) | |
14 | df-ov 6552 | . . . . . 6 ⊢ (𝑥 + 𝑦) = ( + ‘〈𝑥, 𝑦〉) | |
15 | 13, 14 | syl6eqr 2662 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( + ‘𝑝) = (𝑥 + 𝑦)) |
16 | fveq2 6103 | . . . . . 6 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( ⨣ ‘𝑝) = ( ⨣ ‘〈𝑥, 𝑦〉)) | |
17 | df-ov 6552 | . . . . . 6 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘〈𝑥, 𝑦〉) | |
18 | 16, 17 | syl6eqr 2662 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( ⨣ ‘𝑝) = (𝑥 ⨣ 𝑦)) |
19 | 15, 18 | eqeq12d 2625 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))) |
20 | 19 | ralxp 5185 | . . 3 ⊢ (∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) |
21 | 12, 20 | sylibr 223 | . 2 ⊢ (∅ ∉ ran ⨣ → ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) |
22 | fndm 5904 | . . . . 5 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → dom ⨣ = (𝐵 × 𝐵)) | |
23 | 22 | eqcomd 2616 | . . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom ⨣ ) |
24 | 3, 23 | ax-mp 5 | . . 3 ⊢ (𝐵 × 𝐵) = dom ⨣ |
25 | 24 | fveqressseq 6263 | . 2 ⊢ ((Fun ⨣ ∧ ∅ ∉ ran ⨣ ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
26 | 6, 7, 21, 25 | syl3anc 1318 | 1 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 ∀wral 2896 ∅c0 3874 〈cop 4131 × cxp 5036 dom cdm 5038 ran crn 5039 ↾ cres 5040 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 +𝑓cplusf 17062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-plusf 17064 |
This theorem is referenced by: mgmplusfreseq 41563 |
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