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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 | ⊢ 0 = (𝐼 × {0}) |
rrx0el.p | ⊢ 𝑃 = (ℝ ↑m 𝐼) |
Ref | Expression |
---|---|
rrx0el | ⊢ (𝐼 ∈ 𝑉 → 0 ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 11258 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1 | fconst 6788 | . . . . 5 ⊢ (𝐼 × {0}):𝐼⟶{0} |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}):𝐼⟶{0}) |
4 | 0re 11266 | . . . . . 6 ⊢ 0 ∈ ℝ | |
5 | snssg 4792 | . . . . . . 7 ⊢ (0 ∈ ℝ → (0 ∈ ℝ ↔ {0} ⊆ ℝ)) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (0 ∈ ℝ ↔ {0} ⊆ ℝ) |
7 | 4, 6 | mpbi 229 | . . . . 5 ⊢ {0} ⊆ ℝ |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → {0} ⊆ ℝ) |
9 | 3, 8 | fssd 6745 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}):𝐼⟶ℝ) |
10 | reex 11249 | . . . . 5 ⊢ ℝ ∈ V | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝ ∈ V) |
12 | id 22 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉) | |
13 | 11, 12 | elmapd 8869 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((𝐼 × {0}) ∈ (ℝ ↑m 𝐼) ↔ (𝐼 × {0}):𝐼⟶ℝ)) |
14 | 9, 13 | mpbird 256 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ (ℝ ↑m 𝐼)) |
15 | rrx0el.0 | . 2 ⊢ 0 = (𝐼 × {0}) | |
16 | rrx0el.p | . 2 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
17 | 14, 15, 16 | 3eltr4g 2843 | 1 ⊢ (𝐼 ∈ 𝑉 → 0 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3947 {csn 4633 × cxp 5680 ⟶wf 6550 (class class class)co 7424 ↑m cmap 8855 ℝcr 11157 0cc0 11158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-i2m1 11226 ax-rnegex 11229 ax-cnre 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-map 8857 |
This theorem is referenced by: ehl2eudisval0 48113 2sphere0 48138 |
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