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Mirrors > Home > MPE Home > Th. List > pco0 | Structured version Visualization version GIF version |
Description: The starting point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) |
Ref | Expression |
---|---|
pcoval.2 | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
pcoval.3 | ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
Ref | Expression |
---|---|
pco0 | ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘0) = (𝐹‘0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10631 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | 0le0 11726 | . . . 4 ⊢ 0 ≤ 0 | |
3 | halfge0 11842 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
4 | halfre 11839 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
5 | 1, 4 | elicc2i 12790 | . . . 4 ⊢ (0 ∈ (0[,](1 / 2)) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 ≤ (1 / 2))) |
6 | 1, 2, 3, 5 | mpbir3an 1333 | . . 3 ⊢ 0 ∈ (0[,](1 / 2)) |
7 | pcoval.2 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
8 | pcoval.3 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
9 | 7, 8 | pcoval1 23544 | . . 3 ⊢ ((𝜑 ∧ 0 ∈ (0[,](1 / 2))) → ((𝐹(*𝑝‘𝐽)𝐺)‘0) = (𝐹‘(2 · 0))) |
10 | 6, 9 | mpan2 687 | . 2 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘0) = (𝐹‘(2 · 0))) |
11 | 2t0e0 11794 | . . 3 ⊢ (2 · 0) = 0 | |
12 | 11 | fveq2i 6666 | . 2 ⊢ (𝐹‘(2 · 0)) = (𝐹‘0) |
13 | 10, 12 | syl6eq 2869 | 1 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘0) = (𝐹‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 · cmul 10530 ≤ cle 10664 / cdiv 11285 2c2 11680 [,]cicc 12729 Cn ccn 21760 IIcii 23410 *𝑝cpco 23531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-2 11688 df-icc 12733 df-top 21430 df-topon 21447 df-cn 21763 df-pco 23536 |
This theorem is referenced by: pcohtpylem 23550 pcoass 23555 pcorevlem 23557 pcophtb 23560 om1addcl 23564 pi1xfrf 23584 pi1xfr 23586 pi1xfrcnvlem 23587 pi1coghm 23592 connpconn 32379 sconnpht2 32382 cvmlift3lem6 32468 |
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