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Mathbox for Alexander van der Vekens |
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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 17853 | . . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵) |
4 | fnfun 6423 | . . . 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 17851 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦)) |
10 | 9 | eqcomd 2804 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) |
11 | 10 | rgen2 3168 | . . . 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 6645 | . . . . . 6 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( + ‘𝑝) = ( + ‘〈𝑥, 𝑦〉)) | |
14 | df-ov 7138 | . . . . . 6 ⊢ (𝑥 + 𝑦) = ( + ‘〈𝑥, 𝑦〉) | |
15 | 13, 14 | eqtr4di 2851 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( + ‘𝑝) = (𝑥 + 𝑦)) |
16 | fveq2 6645 | . . . . . 6 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( ⨣ ‘𝑝) = ( ⨣ ‘〈𝑥, 𝑦〉)) | |
17 | df-ov 7138 | . . . . . 6 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘〈𝑥, 𝑦〉) | |
18 | 16, 17 | eqtr4di 2851 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( ⨣ ‘𝑝) = (𝑥 ⨣ 𝑦)) |
19 | 15, 18 | eqeq12d 2814 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))) |
20 | 19 | ralxp 5676 | . . 3 ⊢ (∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) |
21 | 12, 20 | sylibr 237 | . 2 ⊢ (∅ ∉ ran ⨣ → ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) |
22 | fndm 6425 | . . . . 5 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → dom ⨣ = (𝐵 × 𝐵)) | |
23 | 22 | eqcomd 2804 | . . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom ⨣ ) |
24 | 3, 23 | ax-mp 5 | . . 3 ⊢ (𝐵 × 𝐵) = dom ⨣ |
25 | 24 | fveqressseq 6824 | . 2 ⊢ ((Fun ⨣ ∧ ∅ ∉ ran ⨣ ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
26 | 6, 7, 21, 25 | syl3anc 1368 | 1 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 ∀wral 3106 ∅c0 4243 〈cop 4531 × cxp 5517 dom cdm 5519 ran crn 5520 ↾ cres 5521 Fun wfun 6318 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 +𝑓cplusf 17841 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-plusf 17843 |
This theorem is referenced by: mgmplusfreseq 44393 |
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