Theorem pcofval 24935

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 7354	. . . 4 ⊢ (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
2		eqidd 2732	. . . 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 7421	. . 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 24930	. . 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 7379	. . . 4 ⊢ (II Cn 𝐽) ∈ V
6	5, 5	mpoex 8011	. . 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 6929	. 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 6960	. . 3 ⊢ (¬ 𝐽 ∈ Top → (*𝑝‘𝐽) = ∅)
9		cntop2 23154	. . . . . . 7 ⊢ (𝑓 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
10	9	con3i 154	. . . . . 6 ⊢ (¬ 𝐽 ∈ Top → ¬ 𝑓 ∈ (II Cn 𝐽))
11	10	eq0rdv 4357	. . . . 5 ⊢ (¬ 𝐽 ∈ Top → (II Cn 𝐽) = ∅)
12	11	olcd 874	. . . 4 ⊢ (¬ 𝐽 ∈ Top → ((II Cn 𝐽) = ∅ ∨ (II Cn 𝐽) = ∅))
13		0mpo0 7429	. . . 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 2769	. 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 1541   ∈ wcel 2111  ∅c0 4283  ifcif 4475   class class class wbr 5091   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  0cc0 11003  1c1 11004   · cmul 11008   ≤ cle 11144   − cmin 11341   / cdiv 11771  2c2 12177  [,]cicc 13245  Topctop 22806   Cn ccn 23137  IIcii 24793  *_𝑝cpco 24925

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-top 22807  df-topon 22824  df-cn 23140  df-pco 24930

This theorem is referenced by:  pcoval  24936