Theorem pcofval 24926

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.)

Assertion

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)))))

Distinct variable group:   𝑓,𝑔,𝑥,𝐽

Proof of Theorem pcofval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7361	. . . 4 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
2		eqidd 2730	. . . 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 7428	. . 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 24921	. . 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 7386	. . . 4 ⊢ (II Cn 𝐽) ∈ V
6	5, 5	mpoex 8021	. . 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 6934	. 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 6965	. . 3 ⊢ (¬ 𝐽 ∈ Top → (*𝑝‘𝐽) = ∅)
9		cntop2 23144	. . . . . . 7 ⊢ (𝑓 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
10	9	con3i 154	. . . . . 6 ⊢ (¬ 𝐽 ∈ Top → ¬ 𝑓 ∈ (II Cn 𝐽))
11	10	eq0rdv 4360	. . . . 5 ⊢ (¬ 𝐽 ∈ Top → (II Cn 𝐽) = ∅)
12	11	olcd 874	. . . 4 ⊢ (¬ 𝐽 ∈ Top → ((II Cn 𝐽) = ∅ ∨ (II Cn 𝐽) = ∅))
13		0mpo0 7436	. . . 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 2767	. 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 182	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)))))

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∅c0 4286  ifcif 4478   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  0cc0 11028  1c1 11029   · cmul 11033   ≤ cle 11169   − cmin 11365   / cdiv 11795  2c2 12201  [,]cicc 13269  Topctop 22796   Cn ccn 23127  IIcii 24784  *_𝑝cpco 24916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-map 8762  df-top 22797  df-topon 22814  df-cn 23130  df-pco 24921

This theorem is referenced by:  pcoval  24927