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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 18663 | . . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵) | 
| 4 | fnfun 6667 | . . . 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 18661 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦)) | 
| 10 | 9 | eqcomd 2742 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) | 
| 11 | 10 | rgen2 3198 | . . . 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 6905 | . . . . . 6 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( + ‘𝑝) = ( + ‘〈𝑥, 𝑦〉)) | |
| 14 | df-ov 7435 | . . . . . 6 ⊢ (𝑥 + 𝑦) = ( + ‘〈𝑥, 𝑦〉) | |
| 15 | 13, 14 | eqtr4di 2794 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( + ‘𝑝) = (𝑥 + 𝑦)) | 
| 16 | fveq2 6905 | . . . . . 6 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( ⨣ ‘𝑝) = ( ⨣ ‘〈𝑥, 𝑦〉)) | |
| 17 | df-ov 7435 | . . . . . 6 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘〈𝑥, 𝑦〉) | |
| 18 | 16, 17 | eqtr4di 2794 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( ⨣ ‘𝑝) = (𝑥 ⨣ 𝑦)) | 
| 19 | 15, 18 | eqeq12d 2752 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))) | 
| 20 | 19 | ralxp 5851 | . . 3 ⊢ (∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) | 
| 21 | 12, 20 | sylibr 234 | . 2 ⊢ (∅ ∉ ran ⨣ → ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) | 
| 22 | fndm 6670 | . . . . 5 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → dom ⨣ = (𝐵 × 𝐵)) | |
| 23 | 22 | eqcomd 2742 | . . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom ⨣ ) | 
| 24 | 3, 23 | ax-mp 5 | . . 3 ⊢ (𝐵 × 𝐵) = dom ⨣ | 
| 25 | 24 | fveqressseq 7098 | . 2 ⊢ ((Fun ⨣ ∧ ∅ ∉ ran ⨣ ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) | 
| 26 | 6, 7, 21, 25 | syl3anc 1372 | 1 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∉ wnel 3045 ∀wral 3060 ∅c0 4332 〈cop 4631 × cxp 5682 dom cdm 5684 ran crn 5685 ↾ cres 5686 Fun wfun 6554 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 +𝑓cplusf 18651 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-plusf 18653 | 
| This theorem is referenced by: mgmplusfreseq 48086 | 
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