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Mirrors > Home > MPE Home > Th. List > rrx0el | Structured version Visualization version GIF version |
Description: The zero ("origin") in a generalized real Euclidean space is an element of its base set. (Contributed by AV, 11-Feb-2023.) |
Ref | Expression |
---|---|
rrx0el.0 | β’ 0 = (πΌ Γ {0}) |
rrx0el.p | β’ π = (β βm πΌ) |
Ref | Expression |
---|---|
rrx0el | β’ (πΌ β π β 0 β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 11209 | . . . . . 6 β’ 0 β V | |
2 | 1 | fconst 6770 | . . . . 5 β’ (πΌ Γ {0}):πΌβΆ{0} |
3 | 2 | a1i 11 | . . . 4 β’ (πΌ β π β (πΌ Γ {0}):πΌβΆ{0}) |
4 | 0re 11217 | . . . . . 6 β’ 0 β β | |
5 | snssg 4782 | . . . . . . 7 β’ (0 β β β (0 β β β {0} β β)) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 β’ (0 β β β {0} β β) |
7 | 4, 6 | mpbi 229 | . . . . 5 β’ {0} β β |
8 | 7 | a1i 11 | . . . 4 β’ (πΌ β π β {0} β β) |
9 | 3, 8 | fssd 6728 | . . 3 β’ (πΌ β π β (πΌ Γ {0}):πΌβΆβ) |
10 | reex 11200 | . . . . 5 β’ β β V | |
11 | 10 | a1i 11 | . . . 4 β’ (πΌ β π β β β V) |
12 | id 22 | . . . 4 β’ (πΌ β π β πΌ β π) | |
13 | 11, 12 | elmapd 8833 | . . 3 β’ (πΌ β π β ((πΌ Γ {0}) β (β βm πΌ) β (πΌ Γ {0}):πΌβΆβ)) |
14 | 9, 13 | mpbird 257 | . 2 β’ (πΌ β π β (πΌ Γ {0}) β (β βm πΌ)) |
15 | rrx0el.0 | . 2 β’ 0 = (πΌ Γ {0}) | |
16 | rrx0el.p | . 2 β’ π = (β βm πΌ) | |
17 | 14, 15, 16 | 3eltr4g 2844 | 1 β’ (πΌ β π β 0 β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 {csn 4623 Γ cxp 5667 βΆwf 6532 (class class class)co 7404 βm cmap 8819 βcr 11108 0cc0 11109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-i2m1 11177 ax-rnegex 11180 ax-cnre 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 |
This theorem is referenced by: ehl2eudisval0 47667 2sphere0 47692 |
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