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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 18363 | . . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵) |
4 | fnfun 6552 | . . . 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 18361 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦)) |
10 | 9 | eqcomd 2739 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) |
11 | 10 | rgen2 3188 | . . . 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 6792 | . . . . . 6 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( + ‘𝑝) = ( + ‘〈𝑥, 𝑦〉)) | |
14 | df-ov 7298 | . . . . . 6 ⊢ (𝑥 + 𝑦) = ( + ‘〈𝑥, 𝑦〉) | |
15 | 13, 14 | eqtr4di 2791 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( + ‘𝑝) = (𝑥 + 𝑦)) |
16 | fveq2 6792 | . . . . . 6 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( ⨣ ‘𝑝) = ( ⨣ ‘〈𝑥, 𝑦〉)) | |
17 | df-ov 7298 | . . . . . 6 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘〈𝑥, 𝑦〉) | |
18 | 16, 17 | eqtr4di 2791 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → ( ⨣ ‘𝑝) = (𝑥 ⨣ 𝑦)) |
19 | 15, 18 | eqeq12d 2749 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦))) |
20 | 19 | ralxp 5754 | . . 3 ⊢ (∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑥 ⨣ 𝑦)) |
21 | 12, 20 | sylibr 233 | . 2 ⊢ (∅ ∉ ran ⨣ → ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) |
22 | fndm 6555 | . . . . 5 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → dom ⨣ = (𝐵 × 𝐵)) | |
23 | 22 | eqcomd 2739 | . . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom ⨣ ) |
24 | 3, 23 | ax-mp 5 | . . 3 ⊢ (𝐵 × 𝐵) = dom ⨣ |
25 | 24 | fveqressseq 6977 | . 2 ⊢ ((Fun ⨣ ∧ ∅ ∉ ran ⨣ ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( + ‘𝑝) = ( ⨣ ‘𝑝)) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
26 | 6, 7, 21, 25 | syl3anc 1369 | 1 ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ∉ wnel 3044 ∀wral 3059 ∅c0 4259 〈cop 4570 × cxp 5589 dom cdm 5591 ran crn 5592 ↾ cres 5593 Fun wfun 6441 Fn wfn 6442 ‘cfv 6447 (class class class)co 7295 Basecbs 16940 +gcplusg 16990 +𝑓cplusf 18351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-1st 7851 df-2nd 7852 df-plusf 18353 |
This theorem is referenced by: mgmplusfreseq 45367 |
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