Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 17250 | . . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵) |
4 | fnfun 5988 | . . . 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 17248 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦)) |
10 | 9 | eqcomd 2628 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) |
11 | 10 | rgen2a 2977 | . . . 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 6191 | . . . . . 6 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( + ‘𝑝) = ( + ‘〈𝑥, 𝑦〉)) | |
14 | df-ov 6653 | . . . . . 6 ⊢ (𝑥 + 𝑦) = ( + ‘〈𝑥, 𝑦〉) | |
15 | 13, 14 | syl6eqr 2674 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( + ‘𝑝) = (𝑥 + 𝑦)) |
16 | fveq2 6191 | . . . . . 6 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( ⨣ ‘𝑝) = ( ⨣ ‘〈𝑥, 𝑦〉)) | |
17 | df-ov 6653 | . . . . . 6 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘〈𝑥, 𝑦〉) | |
18 | 16, 17 | syl6eqr 2674 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( ⨣ ‘𝑝) = (𝑥 ⨣ 𝑦)) |
19 | 15, 18 | eqeq12d 2637 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))) |
20 | 19 | ralxp 5263 | . . 3 ⊢ (∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) |
21 | 12, 20 | sylibr 224 | . 2 ⊢ (∅ ∉ ran ⨣ → ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) |
22 | fndm 5990 | . . . . 5 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → dom ⨣ = (𝐵 × 𝐵)) | |
23 | 22 | eqcomd 2628 | . . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom ⨣ ) |
24 | 3, 23 | ax-mp 5 | . . 3 ⊢ (𝐵 × 𝐵) = dom ⨣ |
25 | 24 | fveqressseq 6355 | . 2 ⊢ ((Fun ⨣ ∧ ∅ ∉ ran ⨣ ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
26 | 6, 7, 21, 25 | syl3anc 1326 | 1 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∉ wnel 2897 ∀wral 2912 ∅c0 3915 〈cop 4183 × cxp 5112 dom cdm 5114 ran crn 5115 ↾ cres 5116 Fun wfun 5882 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 +𝑓cplusf 17239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-plusf 17241 |
This theorem is referenced by: mgmplusfreseq 41773 |
Copyright terms: Public domain | W3C validator |