Theorem pcofval 25057

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 7439	. . . 4 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
2		eqidd 2736	. . . 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 7508	. . 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 25052	. . 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 7464	. . . 4 ⊢ (II Cn 𝐽) ∈ V
6	5, 5	mpoex 8103	. . 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 7016	. 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 7047	. . 3 ⊢ (¬ 𝐽 ∈ Top → (*𝑝‘𝐽) = ∅)
9		cntop2 23265	. . . . . . 7 ⊢ (𝑓 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
10	9	con3i 154	. . . . . 6 ⊢ (¬ 𝐽 ∈ Top → ¬ 𝑓 ∈ (II Cn 𝐽))
11	10	eq0rdv 4413	. . . . 5 ⊢ (¬ 𝐽 ∈ Top → (II Cn 𝐽) = ∅)
12	11	olcd 874	. . . 4 ⊢ (¬ 𝐽 ∈ Top → ((II Cn 𝐽) = ∅ ∨ (II Cn 𝐽) = ∅))
13		0mpo0 7516	. . . 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 2778	. 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 1537   ∈ wcel 2106  ∅c0 4339  ifcif 4531   class class class wbr 5148   ↦ cmpt 5231  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  0cc0 11153  1c1 11154   · cmul 11158   ≤ cle 11294   − cmin 11490   / cdiv 11918  2c2 12319  [,]cicc 13387  Topctop 22915   Cn ccn 23248  IIcii 24915  *_𝑝cpco 25047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-top 22916  df-topon 22933  df-cn 23251  df-pco 25052

This theorem is referenced by:  pcoval  25058