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Mirrors > Home > MPE Home > Th. List > pcofval | Structured version Visualization version GIF version |
Description: The value of the path concatenation function on a topological space. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.) (Proof shortened by AV, 2-Mar-2024.) |
Ref | Expression |
---|---|
pcofval | ⊢ (*𝑝‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7166 | . . . 4 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽)) | |
2 | eqidd 2824 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))))) | |
3 | 1, 1, 2 | mpoeq123dv 7231 | . . 3 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))))) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))))) |
4 | df-pco 23611 | . . 3 ⊢ *𝑝 = (𝑗 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))))) | |
5 | ovex 7191 | . . . 4 ⊢ (II Cn 𝐽) ∈ V | |
6 | 5, 5 | mpoex 7779 | . . 3 ⊢ (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))))) ∈ V |
7 | 3, 4, 6 | fvmpt 6770 | . 2 ⊢ (𝐽 ∈ Top → (*𝑝‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))))) |
8 | 4 | fvmptndm 6800 | . . 3 ⊢ (¬ 𝐽 ∈ Top → (*𝑝‘𝐽) = ∅) |
9 | cntop2 21851 | . . . . . . 7 ⊢ (𝑓 ∈ (II Cn 𝐽) → 𝐽 ∈ Top) | |
10 | 9 | con3i 157 | . . . . . 6 ⊢ (¬ 𝐽 ∈ Top → ¬ 𝑓 ∈ (II Cn 𝐽)) |
11 | 10 | eq0rdv 4359 | . . . . 5 ⊢ (¬ 𝐽 ∈ Top → (II Cn 𝐽) = ∅) |
12 | 11 | olcd 870 | . . . 4 ⊢ (¬ 𝐽 ∈ Top → ((II Cn 𝐽) = ∅ ∨ (II Cn 𝐽) = ∅)) |
13 | 0mpo0 7239 | . . . 4 ⊢ (((II Cn 𝐽) = ∅ ∨ (II Cn 𝐽) = ∅) → (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))))) = ∅) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (¬ 𝐽 ∈ Top → (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))))) = ∅) |
15 | 8, 14 | eqtr4d 2861 | . 2 ⊢ (¬ 𝐽 ∈ Top → (*𝑝‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))))) |
16 | 7, 15 | pm2.61i 184 | 1 ⊢ (*𝑝‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∅c0 4293 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 0cc0 10539 1c1 10540 · cmul 10544 ≤ cle 10678 − cmin 10872 / cdiv 11299 2c2 11695 [,]cicc 12744 Topctop 21503 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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3145 df-rex 3146 df-reu 3147 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-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-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-top 21504 df-topon 21521 df-cn 21837 df-pco 23611 |
This theorem is referenced by: pcoval 23617 |
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