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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 10645 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | 0le0 11741 | . . . 4 ⊢ 0 ≤ 0 | |
3 | halfge0 11857 | . . . 4 ⊢ 0 ≤ (1 / 2) | |
4 | halfre 11854 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
5 | 1, 4 | elicc2i 12805 | . . . 4 ⊢ (0 ∈ (0[,](1 / 2)) ↔ (0 ∈ ℝ ∧ 0 ≤ 0 ∧ 0 ≤ (1 / 2))) |
6 | 1, 2, 3, 5 | mpbir3an 1337 | . . 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 23619 | . . 3 ⊢ ((𝜑 ∧ 0 ∈ (0[,](1 / 2))) → ((𝐹(*𝑝‘𝐽)𝐺)‘0) = (𝐹‘(2 · 0))) |
10 | 6, 9 | mpan2 689 | . 2 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘0) = (𝐹‘(2 · 0))) |
11 | 2t0e0 11809 | . . 3 ⊢ (2 · 0) = 0 | |
12 | 11 | fveq2i 6675 | . 2 ⊢ (𝐹‘(2 · 0)) = (𝐹‘0) |
13 | 10, 12 | syl6eq 2874 | 1 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘0) = (𝐹‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 · cmul 10544 ≤ cle 10678 / cdiv 11299 2c2 11695 [,]cicc 12744 Cn ccn 21834 IIcii 23485 *𝑝cpco 23606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-icc 12748 df-top 21504 df-topon 21521 df-cn 21837 df-pco 23611 |
This theorem is referenced by: pcohtpylem 23625 pcoass 23630 pcorevlem 23632 pcophtb 23635 om1addcl 23639 pi1xfrf 23659 pi1xfr 23661 pi1xfrcnvlem 23662 pi1coghm 23667 connpconn 32484 sconnpht2 32487 cvmlift3lem6 32573 |
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