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Theorem cnmpopc 23543
 Description: Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
Hypotheses
Ref Expression
cnmpopc.r 𝑅 = (topGen‘ran (,))
cnmpopc.m 𝑀 = (𝑅t (𝐴[,]𝐵))
cnmpopc.n 𝑁 = (𝑅t (𝐵[,]𝐶))
cnmpopc.o 𝑂 = (𝑅t (𝐴[,]𝐶))
cnmpopc.a (𝜑𝐴 ∈ ℝ)
cnmpopc.c (𝜑𝐶 ∈ ℝ)
cnmpopc.b (𝜑𝐵 ∈ (𝐴[,]𝐶))
cnmpopc.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpopc.q ((𝜑 ∧ (𝑥 = 𝐵𝑦𝑋)) → 𝐷 = 𝐸)
cnmpopc.d (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))
cnmpopc.e (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))
Assertion
Ref Expression
cnmpopc (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝑀(𝑥,𝑦)   𝑁(𝑥,𝑦)   𝑂(𝑥,𝑦)

Proof of Theorem cnmpopc
StepHypRef Expression
1 eqid 2798 . 2 (𝑂 ×t 𝐽) = (𝑂 ×t 𝐽)
2 eqid 2798 . 2 𝐾 = 𝐾
3 cnmpopc.a . . . . . 6 (𝜑𝐴 ∈ ℝ)
4 cnmpopc.c . . . . . 6 (𝜑𝐶 ∈ ℝ)
5 iccssre 12810 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴[,]𝐶) ⊆ ℝ)
63, 4, 5syl2anc 587 . . . . 5 (𝜑 → (𝐴[,]𝐶) ⊆ ℝ)
7 cnmpopc.b . . . . . . . 8 (𝜑𝐵 ∈ (𝐴[,]𝐶))
86, 7sseldd 3916 . . . . . . 7 (𝜑𝐵 ∈ ℝ)
9 icccld 23382 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
103, 8, 9syl2anc 587 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
11 cnmpopc.r . . . . . . 7 𝑅 = (topGen‘ran (,))
1211fveq2i 6649 . . . . . 6 (Clsd‘𝑅) = (Clsd‘(topGen‘ran (,)))
1310, 12eleqtrrdi 2901 . . . . 5 (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑅))
14 ssun1 4099 . . . . . 6 (𝐴[,]𝐵) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))
15 iccsplit 12866 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ (𝐴[,]𝐶)) → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)))
163, 4, 7, 15syl3anc 1368 . . . . . 6 (𝜑 → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)))
1714, 16sseqtrrid 3968 . . . . 5 (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶))
18 uniretop 23378 . . . . . . 7 ℝ = (topGen‘ran (,))
1911unieqi 4814 . . . . . . 7 𝑅 = (topGen‘ran (,))
2018, 19eqtr4i 2824 . . . . . 6 ℝ = 𝑅
2120restcldi 21788 . . . . 5 (((𝐴[,]𝐶) ⊆ ℝ ∧ (𝐴[,]𝐵) ∈ (Clsd‘𝑅) ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶)) → (𝐴[,]𝐵) ∈ (Clsd‘(𝑅t (𝐴[,]𝐶))))
226, 13, 17, 21syl3anc 1368 . . . 4 (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(𝑅t (𝐴[,]𝐶))))
23 cnmpopc.o . . . . 5 𝑂 = (𝑅t (𝐴[,]𝐶))
2423fveq2i 6649 . . . 4 (Clsd‘𝑂) = (Clsd‘(𝑅t (𝐴[,]𝐶)))
2522, 24eleqtrrdi 2901 . . 3 (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑂))
26 cnmpopc.j . . . . 5 (𝜑𝐽 ∈ (TopOn‘𝑋))
27 toponuni 21529 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2826, 27syl 17 . . . 4 (𝜑𝑋 = 𝐽)
29 topontop 21528 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
30 eqid 2798 . . . . . 6 𝐽 = 𝐽
3130topcld 21650 . . . . 5 (𝐽 ∈ Top → 𝐽 ∈ (Clsd‘𝐽))
3226, 29, 313syl 18 . . . 4 (𝜑 𝐽 ∈ (Clsd‘𝐽))
3328, 32eqeltrd 2890 . . 3 (𝜑𝑋 ∈ (Clsd‘𝐽))
34 txcld 22218 . . 3 (((𝐴[,]𝐵) ∈ (Clsd‘𝑂) ∧ 𝑋 ∈ (Clsd‘𝐽)) → ((𝐴[,]𝐵) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
3525, 33, 34syl2anc 587 . 2 (𝜑 → ((𝐴[,]𝐵) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
36 icccld 23382 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran (,))))
378, 4, 36syl2anc 587 . . . . . 6 (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran (,))))
3837, 12eleqtrrdi 2901 . . . . 5 (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑅))
39 ssun2 4100 . . . . . 6 (𝐵[,]𝐶) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))
4039, 16sseqtrrid 3968 . . . . 5 (𝜑 → (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶))
4120restcldi 21788 . . . . 5 (((𝐴[,]𝐶) ⊆ ℝ ∧ (𝐵[,]𝐶) ∈ (Clsd‘𝑅) ∧ (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶)) → (𝐵[,]𝐶) ∈ (Clsd‘(𝑅t (𝐴[,]𝐶))))
426, 38, 40, 41syl3anc 1368 . . . 4 (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(𝑅t (𝐴[,]𝐶))))
4342, 24eleqtrrdi 2901 . . 3 (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑂))
44 txcld 22218 . . 3 (((𝐵[,]𝐶) ∈ (Clsd‘𝑂) ∧ 𝑋 ∈ (Clsd‘𝐽)) → ((𝐵[,]𝐶) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
4543, 33, 44syl2anc 587 . 2 (𝜑 → ((𝐵[,]𝐶) × 𝑋) ∈ (Clsd‘(𝑂 ×t 𝐽)))
4616xpeq1d 5549 . . . 4 (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋))
47 xpundir 5586 . . . 4 (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋))
4846, 47eqtrdi 2849 . . 3 (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)))
49 retopon 23379 . . . . . . . 8 (topGen‘ran (,)) ∈ (TopOn‘ℝ)
5011, 49eqeltri 2886 . . . . . . 7 𝑅 ∈ (TopOn‘ℝ)
51 resttopon 21776 . . . . . . 7 ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐴[,]𝐶) ⊆ ℝ) → (𝑅t (𝐴[,]𝐶)) ∈ (TopOn‘(𝐴[,]𝐶)))
5250, 6, 51sylancr 590 . . . . . 6 (𝜑 → (𝑅t (𝐴[,]𝐶)) ∈ (TopOn‘(𝐴[,]𝐶)))
5323, 52eqeltrid 2894 . . . . 5 (𝜑𝑂 ∈ (TopOn‘(𝐴[,]𝐶)))
54 txtopon 22206 . . . . 5 ((𝑂 ∈ (TopOn‘(𝐴[,]𝐶)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)))
5553, 26, 54syl2anc 587 . . . 4 (𝜑 → (𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)))
56 toponuni 21529 . . . 4 ((𝑂 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐶) × 𝑋)) → ((𝐴[,]𝐶) × 𝑋) = (𝑂 ×t 𝐽))
5755, 56syl 17 . . 3 (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (𝑂 ×t 𝐽))
5848, 57eqtr3d 2835 . 2 (𝜑 → (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)) = (𝑂 ×t 𝐽))
59 cnmpopc.m . . . . . . . . . 10 𝑀 = (𝑅t (𝐴[,]𝐵))
6017, 6sstrd 3925 . . . . . . . . . . 11 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
61 resttopon 21776 . . . . . . . . . . 11 ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝑅t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
6250, 60, 61sylancr 590 . . . . . . . . . 10 (𝜑 → (𝑅t (𝐴[,]𝐵)) ∈ (TopOn‘(𝐴[,]𝐵)))
6359, 62eqeltrid 2894 . . . . . . . . 9 (𝜑𝑀 ∈ (TopOn‘(𝐴[,]𝐵)))
64 txtopon 22206 . . . . . . . . 9 ((𝑀 ∈ (TopOn‘(𝐴[,]𝐵)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋)))
6563, 26, 64syl2anc 587 . . . . . . . 8 (𝜑 → (𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋)))
66 cnmpopc.d . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))
67 cntop2 21856 . . . . . . . . . 10 ((𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾) → 𝐾 ∈ Top)
6866, 67syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Top)
69 toptopon2 21533 . . . . . . . . 9 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
7068, 69sylib 221 . . . . . . . 8 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
71 elicc2 12793 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
723, 8, 71syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
7372biimpa 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵))
7473simp3d 1141 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
75743adant3 1129 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦𝑋) → 𝑥𝐵)
7675iftrued 4433 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦𝑋) → if(𝑥𝐵, 𝐷, 𝐸) = 𝐷)
7776mpoeq3dva 7211 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋𝐷))
7877, 66eqeltrd 2890 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))
79 cnf2 21864 . . . . . . . 8 (((𝑀 ×t 𝐽) ∈ (TopOn‘((𝐴[,]𝐵) × 𝑋)) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ∈ ((𝑀 ×t 𝐽) Cn 𝐾)) → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶ 𝐾)
8065, 70, 78, 79syl3anc 1368 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶ 𝐾)
81 eqid 2798 . . . . . . . 8 (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸))
8281fmpo 7751 . . . . . . 7 (∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾 ↔ (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶ 𝐾)
8380, 82sylibr 237 . . . . . 6 (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾)
84 cnmpopc.n . . . . . . . . . 10 𝑁 = (𝑅t (𝐵[,]𝐶))
8540, 6sstrd 3925 . . . . . . . . . . 11 (𝜑 → (𝐵[,]𝐶) ⊆ ℝ)
86 resttopon 21776 . . . . . . . . . . 11 ((𝑅 ∈ (TopOn‘ℝ) ∧ (𝐵[,]𝐶) ⊆ ℝ) → (𝑅t (𝐵[,]𝐶)) ∈ (TopOn‘(𝐵[,]𝐶)))
8750, 85, 86sylancr 590 . . . . . . . . . 10 (𝜑 → (𝑅t (𝐵[,]𝐶)) ∈ (TopOn‘(𝐵[,]𝐶)))
8884, 87eqeltrid 2894 . . . . . . . . 9 (𝜑𝑁 ∈ (TopOn‘(𝐵[,]𝐶)))
89 txtopon 22206 . . . . . . . . 9 ((𝑁 ∈ (TopOn‘(𝐵[,]𝐶)) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)))
9088, 26, 89syl2anc 587 . . . . . . . 8 (𝜑 → (𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)))
91 elicc2 12793 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝑥 ∈ (𝐵[,]𝐶) ↔ (𝑥 ∈ ℝ ∧ 𝐵𝑥𝑥𝐶)))
928, 4, 91syl2anc 587 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑥 ∈ (𝐵[,]𝐶) ↔ (𝑥 ∈ ℝ ∧ 𝐵𝑥𝑥𝐶)))
9392biimpa 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 ∈ ℝ ∧ 𝐵𝑥𝑥𝐶))
9493simp2d 1140 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → 𝐵𝑥)
9594biantrud 535 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → (𝑥𝐵 ↔ (𝑥𝐵𝐵𝑥)))
9693simp1d 1139 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → 𝑥 ∈ ℝ)
978adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ∈ ℝ)
9896, 97letri3d 10774 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 = 𝐵 ↔ (𝑥𝐵𝐵𝑥)))
9995, 98bitr4d 285 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → (𝑥𝐵𝑥 = 𝐵))
100993adant3 1129 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦𝑋) → (𝑥𝐵𝑥 = 𝐵))
101 cnmpopc.q . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 = 𝐵𝑦𝑋)) → 𝐷 = 𝐸)
102101ancom2s 649 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦𝑋𝑥 = 𝐵)) → 𝐷 = 𝐸)
103102ifeq1d 4443 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝑋𝑥 = 𝐵)) → if(𝑥𝐵, 𝐷, 𝐸) = if(𝑥𝐵, 𝐸, 𝐸))
104 ifid 4464 . . . . . . . . . . . . . . 15 if(𝑥𝐵, 𝐸, 𝐸) = 𝐸
105103, 104eqtrdi 2849 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑋𝑥 = 𝐵)) → if(𝑥𝐵, 𝐷, 𝐸) = 𝐸)
106105expr 460 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋) → (𝑥 = 𝐵 → if(𝑥𝐵, 𝐷, 𝐸) = 𝐸))
1071063adant2 1128 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦𝑋) → (𝑥 = 𝐵 → if(𝑥𝐵, 𝐷, 𝐸) = 𝐸))
108100, 107sylbid 243 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦𝑋) → (𝑥𝐵 → if(𝑥𝐵, 𝐷, 𝐸) = 𝐸))
109 iffalse 4434 . . . . . . . . . . 11 𝑥𝐵 → if(𝑥𝐵, 𝐷, 𝐸) = 𝐸)
110108, 109pm2.61d1 183 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦𝑋) → if(𝑥𝐵, 𝐷, 𝐸) = 𝐸)
111110mpoeq3dva 7211 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋𝐸))
112 cnmpopc.e . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))
113111, 112eqeltrd 2890 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))
114 cnf2 21864 . . . . . . . 8 (((𝑁 ×t 𝐽) ∈ (TopOn‘((𝐵[,]𝐶) × 𝑋)) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ∈ ((𝑁 ×t 𝐽) Cn 𝐾)) → (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶ 𝐾)
11590, 70, 113, 114syl3anc 1368 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶ 𝐾)
116 eqid 2798 . . . . . . . 8 (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸))
117116fmpo 7751 . . . . . . 7 (∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾 ↔ (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶ 𝐾)
118115, 117sylibr 237 . . . . . 6 (𝜑 → ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾)
119 ralun 4119 . . . . . 6 ((∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾 ∧ ∀𝑥 ∈ (𝐵[,]𝐶)∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾) → ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾)
12083, 118, 119syl2anc 587 . . . . 5 (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾)
12116raleqdv 3364 . . . . 5 (𝜑 → (∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾 ↔ ∀𝑥 ∈ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾))
122120, 121mpbird 260 . . . 4 (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾)
123 eqid 2798 . . . . 5 (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸))
124123fmpo 7751 . . . 4 (∀𝑥 ∈ (𝐴[,]𝐶)∀𝑦𝑋 if(𝑥𝐵, 𝐷, 𝐸) ∈ 𝐾 ↔ (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶ 𝐾)
125122, 124sylib 221 . . 3 (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶ 𝐾)
12657feq2d 6474 . . 3 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐴[,]𝐶) × 𝑋)⟶ 𝐾 ↔ (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)): (𝑂 ×t 𝐽)⟶ 𝐾))
127125, 126mpbid 235 . 2 (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)): (𝑂 ×t 𝐽)⟶ 𝐾)
128 ssid 3937 . . . 4 𝑋𝑋
129 resmpo 7252 . . . 4 (((𝐴[,]𝐵) ⊆ (𝐴[,]𝐶) ∧ 𝑋𝑋) → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)))
13017, 128, 129sylancl 589 . . 3 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)))
131 retop 23377 . . . . . . . . . 10 (topGen‘ran (,)) ∈ Top
13211, 131eqeltri 2886 . . . . . . . . 9 𝑅 ∈ Top
133 ovex 7169 . . . . . . . . 9 (𝐴[,]𝐶) ∈ V
134 resttop 21775 . . . . . . . . 9 ((𝑅 ∈ Top ∧ (𝐴[,]𝐶) ∈ V) → (𝑅t (𝐴[,]𝐶)) ∈ Top)
135132, 133, 134mp2an 691 . . . . . . . 8 (𝑅t (𝐴[,]𝐶)) ∈ Top
13623, 135eqeltri 2886 . . . . . . 7 𝑂 ∈ Top
137136a1i 11 . . . . . 6 (𝜑𝑂 ∈ Top)
138 ovexd 7171 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ∈ V)
139 txrest 22246 . . . . . 6 (((𝑂 ∈ Top ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ ((𝐴[,]𝐵) ∈ V ∧ 𝑋 ∈ (Clsd‘𝐽))) → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = ((𝑂t (𝐴[,]𝐵)) ×t (𝐽t 𝑋)))
140137, 26, 138, 33, 139syl22anc 837 . . . . 5 (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = ((𝑂t (𝐴[,]𝐵)) ×t (𝐽t 𝑋)))
141132a1i 11 . . . . . . . 8 (𝜑𝑅 ∈ Top)
142 ovexd 7171 . . . . . . . 8 (𝜑 → (𝐴[,]𝐶) ∈ V)
143 restabs 21780 . . . . . . . 8 ((𝑅 ∈ Top ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅t (𝐴[,]𝐵)))
144141, 17, 142, 143syl3anc 1368 . . . . . . 7 (𝜑 → ((𝑅t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅t (𝐴[,]𝐵)))
14523oveq1i 7146 . . . . . . 7 (𝑂t (𝐴[,]𝐵)) = ((𝑅t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵))
146144, 145, 593eqtr4g 2858 . . . . . 6 (𝜑 → (𝑂t (𝐴[,]𝐵)) = 𝑀)
14728oveq2d 7152 . . . . . . 7 (𝜑 → (𝐽t 𝑋) = (𝐽t 𝐽))
14830restid 16702 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → (𝐽t 𝐽) = 𝐽)
14926, 148syl 17 . . . . . . 7 (𝜑 → (𝐽t 𝐽) = 𝐽)
150147, 149eqtrd 2833 . . . . . 6 (𝜑 → (𝐽t 𝑋) = 𝐽)
151146, 150oveq12d 7154 . . . . 5 (𝜑 → ((𝑂t (𝐴[,]𝐵)) ×t (𝐽t 𝑋)) = (𝑀 ×t 𝐽))
152140, 151eqtrd 2833 . . . 4 (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = (𝑀 ×t 𝐽))
153152oveq1d 7151 . . 3 (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾) = ((𝑀 ×t 𝐽) Cn 𝐾))
15478, 130, 1533eltr4d 2905 . 2 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾))
155 resmpo 7252 . . . 4 (((𝐵[,]𝐶) ⊆ (𝐴[,]𝐶) ∧ 𝑋𝑋) → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)))
15640, 128, 155sylancl 589 . . 3 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)))
157 ovexd 7171 . . . . . 6 (𝜑 → (𝐵[,]𝐶) ∈ V)
158 txrest 22246 . . . . . 6 (((𝑂 ∈ Top ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ ((𝐵[,]𝐶) ∈ V ∧ 𝑋 ∈ (Clsd‘𝐽))) → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = ((𝑂t (𝐵[,]𝐶)) ×t (𝐽t 𝑋)))
159137, 26, 157, 33, 158syl22anc 837 . . . . 5 (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = ((𝑂t (𝐵[,]𝐶)) ×t (𝐽t 𝑋)))
160 restabs 21780 . . . . . . . 8 ((𝑅 ∈ Top ∧ (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅t (𝐵[,]𝐶)))
161141, 40, 142, 160syl3anc 1368 . . . . . . 7 (𝜑 → ((𝑅t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅t (𝐵[,]𝐶)))
16223oveq1i 7146 . . . . . . 7 (𝑂t (𝐵[,]𝐶)) = ((𝑅t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶))
163161, 162, 843eqtr4g 2858 . . . . . 6 (𝜑 → (𝑂t (𝐵[,]𝐶)) = 𝑁)
164163, 150oveq12d 7154 . . . . 5 (𝜑 → ((𝑂t (𝐵[,]𝐶)) ×t (𝐽t 𝑋)) = (𝑁 ×t 𝐽))
165159, 164eqtrd 2833 . . . 4 (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = (𝑁 ×t 𝐽))
166165oveq1d 7151 . . 3 (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾) = ((𝑁 ×t 𝐽) Cn 𝐾))
167113, 156, 1663eltr4d 2905 . 2 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ↾ ((𝐵[,]𝐶) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾))
1681, 2, 35, 45, 58, 127, 154, 167paste 21909 1 (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  ifcif 4425  ∪ cuni 4801   class class class wbr 5031   × cxp 5518  ran crn 5521   ↾ cres 5522  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  ℝcr 10528   ≤ cle 10668  (,)cioo 12729  [,]cicc 12732   ↾t crest 16689  topGenctg 16706  Topctop 21508  TopOnctopon 21525  Clsdccld 21631   Cn ccn 21839   ×t ctx 22175 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-oadd 8092  df-er 8275  df-map 8394  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-fi 8862  df-sup 8893  df-inf 8894  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11629  df-n0 11889  df-z 11973  df-uz 12235  df-q 12340  df-ioo 12733  df-icc 12736  df-rest 16691  df-topgen 16712  df-top 21509  df-topon 21526  df-bases 21561  df-cld 21634  df-cn 21842  df-tx 22177 This theorem is referenced by:  htpycc  23595  pcocn  23632  pcohtpylem  23634  pcopt  23637  pcopt2  23638  pcoass  23639  pcorevlem  23641
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