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Theorem cnmpopc 23825
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 2737 . 2 (𝑂 ×t 𝐽) = (𝑂 ×t 𝐽)
2 eqid 2737 . 2 𝐾 = 𝐾
3 cnmpopc.a . . . . . 6 (𝜑𝐴 ∈ ℝ)
4 cnmpopc.c . . . . . 6 (𝜑𝐶 ∈ ℝ)
5 iccssre 13017 . . . . . 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 3902 . . . . . . 7 (𝜑𝐵 ∈ ℝ)
9 icccld 23664 . . . . . . 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 6720 . . . . . 6 (Clsd‘𝑅) = (Clsd‘(topGen‘ran (,)))
1310, 12eleqtrrdi 2849 . . . . 5 (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑅))
14 ssun1 4086 . . . . . 6 (𝐴[,]𝐵) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))
15 iccsplit 13073 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ (𝐴[,]𝐶)) → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)))
163, 4, 7, 15syl3anc 1373 . . . . . 6 (𝜑 → (𝐴[,]𝐶) = ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)))
1714, 16sseqtrrid 3954 . . . . 5 (𝜑 → (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶))
18 uniretop 23660 . . . . . . 7 ℝ = (topGen‘ran (,))
1911unieqi 4832 . . . . . . 7 𝑅 = (topGen‘ran (,))
2018, 19eqtr4i 2768 . . . . . 6 ℝ = 𝑅
2120restcldi 22070 . . . . 5 (((𝐴[,]𝐶) ⊆ ℝ ∧ (𝐴[,]𝐵) ∈ (Clsd‘𝑅) ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶)) → (𝐴[,]𝐵) ∈ (Clsd‘(𝑅t (𝐴[,]𝐶))))
226, 13, 17, 21syl3anc 1373 . . . 4 (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘(𝑅t (𝐴[,]𝐶))))
23 cnmpopc.o . . . . 5 𝑂 = (𝑅t (𝐴[,]𝐶))
2423fveq2i 6720 . . . 4 (Clsd‘𝑂) = (Clsd‘(𝑅t (𝐴[,]𝐶)))
2522, 24eleqtrrdi 2849 . . 3 (𝜑 → (𝐴[,]𝐵) ∈ (Clsd‘𝑂))
26 cnmpopc.j . . . . 5 (𝜑𝐽 ∈ (TopOn‘𝑋))
27 toponuni 21811 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2826, 27syl 17 . . . 4 (𝜑𝑋 = 𝐽)
29 topontop 21810 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
30 eqid 2737 . . . . . 6 𝐽 = 𝐽
3130topcld 21932 . . . . 5 (𝐽 ∈ Top → 𝐽 ∈ (Clsd‘𝐽))
3226, 29, 313syl 18 . . . 4 (𝜑 𝐽 ∈ (Clsd‘𝐽))
3328, 32eqeltrd 2838 . . 3 (𝜑𝑋 ∈ (Clsd‘𝐽))
34 txcld 22500 . . 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 23664 . . . . . . 7 ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran (,))))
378, 4, 36syl2anc 587 . . . . . 6 (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(topGen‘ran (,))))
3837, 12eleqtrrdi 2849 . . . . 5 (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑅))
39 ssun2 4087 . . . . . 6 (𝐵[,]𝐶) ⊆ ((𝐴[,]𝐵) ∪ (𝐵[,]𝐶))
4039, 16sseqtrrid 3954 . . . . 5 (𝜑 → (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶))
4120restcldi 22070 . . . . 5 (((𝐴[,]𝐶) ⊆ ℝ ∧ (𝐵[,]𝐶) ∈ (Clsd‘𝑅) ∧ (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶)) → (𝐵[,]𝐶) ∈ (Clsd‘(𝑅t (𝐴[,]𝐶))))
426, 38, 40, 41syl3anc 1373 . . . 4 (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘(𝑅t (𝐴[,]𝐶))))
4342, 24eleqtrrdi 2849 . . 3 (𝜑 → (𝐵[,]𝐶) ∈ (Clsd‘𝑂))
44 txcld 22500 . . 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 5580 . . . 4 (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋))
47 xpundir 5618 . . . 4 (((𝐴[,]𝐵) ∪ (𝐵[,]𝐶)) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋))
4846, 47eqtrdi 2794 . . 3 (𝜑 → ((𝐴[,]𝐶) × 𝑋) = (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)))
49 retopon 23661 . . . . . . . 8 (topGen‘ran (,)) ∈ (TopOn‘ℝ)
5011, 49eqeltri 2834 . . . . . . 7 𝑅 ∈ (TopOn‘ℝ)
51 resttopon 22058 . . . . . . 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 2842 . . . . 5 (𝜑𝑂 ∈ (TopOn‘(𝐴[,]𝐶)))
54 txtopon 22488 . . . . 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 21811 . . . 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 2779 . 2 (𝜑 → (((𝐴[,]𝐵) × 𝑋) ∪ ((𝐵[,]𝐶) × 𝑋)) = (𝑂 ×t 𝐽))
59 cnmpopc.m . . . . . . . . . 10 𝑀 = (𝑅t (𝐴[,]𝐵))
6017, 6sstrd 3911 . . . . . . . . . . 11 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
61 resttopon 22058 . . . . . . . . . . 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 2842 . . . . . . . . 9 (𝜑𝑀 ∈ (TopOn‘(𝐴[,]𝐵)))
64 txtopon 22488 . . . . . . . . 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 22138 . . . . . . . . . 10 ((𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾) → 𝐾 ∈ Top)
6866, 67syl 17 . . . . . . . . 9 (𝜑𝐾 ∈ Top)
69 toptopon2 21815 . . . . . . . . 9 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
7068, 69sylib 221 . . . . . . . 8 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
71 elicc2 13000 . . . . . . . . . . . . . . 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 1146 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
75743adant3 1134 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦𝑋) → 𝑥𝐵)
7675iftrued 4447 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦𝑋) → if(𝑥𝐵, 𝐷, 𝐸) = 𝐷)
7776mpoeq3dva 7288 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋𝐷))
7877, 66eqeltrd 2838 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))
79 cnf2 22146 . . . . . . . 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 1373 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐴[,]𝐵) × 𝑋)⟶ 𝐾)
81 eqid 2737 . . . . . . . 8 (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸))
8281fmpo 7838 . . . . . . 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 3911 . . . . . . . . . . 11 (𝜑 → (𝐵[,]𝐶) ⊆ ℝ)
86 resttopon 22058 . . . . . . . . . . 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 2842 . . . . . . . . 9 (𝜑𝑁 ∈ (TopOn‘(𝐵[,]𝐶)))
89 txtopon 22488 . . . . . . . . 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 13000 . . . . . . . . . . . . . . . . . 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 1145 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → 𝐵𝑥)
9594biantrud 535 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → (𝑥𝐵 ↔ (𝑥𝐵𝐵𝑥)))
9693simp1d 1144 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → 𝑥 ∈ ℝ)
978adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ∈ ℝ)
9896, 97letri3d 10974 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → (𝑥 = 𝐵 ↔ (𝑥𝐵𝐵𝑥)))
9995, 98bitr4d 285 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐵[,]𝐶)) → (𝑥𝐵𝑥 = 𝐵))
100993adant3 1134 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦𝑋) → (𝑥𝐵𝑥 = 𝐵))
101 cnmpopc.q . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑥 = 𝐵𝑦𝑋)) → 𝐷 = 𝐸)
102101ancom2s 650 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦𝑋𝑥 = 𝐵)) → 𝐷 = 𝐸)
103102ifeq1d 4458 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑦𝑋𝑥 = 𝐵)) → if(𝑥𝐵, 𝐷, 𝐸) = if(𝑥𝐵, 𝐸, 𝐸))
104 ifid 4479 . . . . . . . . . . . . . . 15 if(𝑥𝐵, 𝐸, 𝐸) = 𝐸
105103, 104eqtrdi 2794 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑦𝑋𝑥 = 𝐵)) → if(𝑥𝐵, 𝐷, 𝐸) = 𝐸)
106105expr 460 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋) → (𝑥 = 𝐵 → if(𝑥𝐵, 𝐷, 𝐸) = 𝐸))
1071063adant2 1133 . . . . . . . . . . . 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 4448 . . . . . . . . . . 11 𝑥𝐵 → if(𝑥𝐵, 𝐷, 𝐸) = 𝐸)
110108, 109pm2.61d1 183 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐵[,]𝐶) ∧ 𝑦𝑋) → if(𝑥𝐵, 𝐷, 𝐸) = 𝐸)
111110mpoeq3dva 7288 . . . . . . . . 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 2838 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))
114 cnf2 22146 . . . . . . . 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 1373 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)):((𝐵[,]𝐶) × 𝑋)⟶ 𝐾)
116 eqid 2737 . . . . . . . 8 (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸))
117116fmpo 7838 . . . . . . 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 4106 . . . . . 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 3325 . . . . 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 2737 . . . . 5 (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) = (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸))
124123fmpo 7838 . . . 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 6531 . . 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 3923 . . . 4 𝑋𝑋
129 resmpo 7330 . . . 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 23659 . . . . . . . . . 10 (topGen‘ran (,)) ∈ Top
13211, 131eqeltri 2834 . . . . . . . . 9 𝑅 ∈ Top
133 ovex 7246 . . . . . . . . 9 (𝐴[,]𝐶) ∈ V
134 resttop 22057 . . . . . . . . 9 ((𝑅 ∈ Top ∧ (𝐴[,]𝐶) ∈ V) → (𝑅t (𝐴[,]𝐶)) ∈ Top)
135132, 133, 134mp2an 692 . . . . . . . 8 (𝑅t (𝐴[,]𝐶)) ∈ Top
13623, 135eqeltri 2834 . . . . . . 7 𝑂 ∈ Top
137136a1i 11 . . . . . 6 (𝜑𝑂 ∈ Top)
138 ovexd 7248 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ∈ V)
139 txrest 22528 . . . . . 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 839 . . . . 5 (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = ((𝑂t (𝐴[,]𝐵)) ×t (𝐽t 𝑋)))
141132a1i 11 . . . . . . . 8 (𝜑𝑅 ∈ Top)
142 ovexd 7248 . . . . . . . 8 (𝜑 → (𝐴[,]𝐶) ∈ V)
143 restabs 22062 . . . . . . . 8 ((𝑅 ∈ Top ∧ (𝐴[,]𝐵) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅t (𝐴[,]𝐵)))
144141, 17, 142, 143syl3anc 1373 . . . . . . 7 (𝜑 → ((𝑅t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵)) = (𝑅t (𝐴[,]𝐵)))
14523oveq1i 7223 . . . . . . 7 (𝑂t (𝐴[,]𝐵)) = ((𝑅t (𝐴[,]𝐶)) ↾t (𝐴[,]𝐵))
146144, 145, 593eqtr4g 2803 . . . . . 6 (𝜑 → (𝑂t (𝐴[,]𝐵)) = 𝑀)
14728oveq2d 7229 . . . . . . 7 (𝜑 → (𝐽t 𝑋) = (𝐽t 𝐽))
14830restid 16938 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → (𝐽t 𝐽) = 𝐽)
14926, 148syl 17 . . . . . . 7 (𝜑 → (𝐽t 𝐽) = 𝐽)
150147, 149eqtrd 2777 . . . . . 6 (𝜑 → (𝐽t 𝑋) = 𝐽)
151146, 150oveq12d 7231 . . . . 5 (𝜑 → ((𝑂t (𝐴[,]𝐵)) ×t (𝐽t 𝑋)) = (𝑀 ×t 𝐽))
152140, 151eqtrd 2777 . . . 4 (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) = (𝑀 ×t 𝐽))
153152oveq1d 7228 . . 3 (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾) = ((𝑀 ×t 𝐽) Cn 𝐾))
15478, 130, 1533eltr4d 2853 . 2 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ↾ ((𝐴[,]𝐵) × 𝑋)) ∈ (((𝑂 ×t 𝐽) ↾t ((𝐴[,]𝐵) × 𝑋)) Cn 𝐾))
155 resmpo 7330 . . . 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 7248 . . . . . 6 (𝜑 → (𝐵[,]𝐶) ∈ V)
158 txrest 22528 . . . . . 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 839 . . . . 5 (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = ((𝑂t (𝐵[,]𝐶)) ×t (𝐽t 𝑋)))
160 restabs 22062 . . . . . . . 8 ((𝑅 ∈ Top ∧ (𝐵[,]𝐶) ⊆ (𝐴[,]𝐶) ∧ (𝐴[,]𝐶) ∈ V) → ((𝑅t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅t (𝐵[,]𝐶)))
161141, 40, 142, 160syl3anc 1373 . . . . . . 7 (𝜑 → ((𝑅t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶)) = (𝑅t (𝐵[,]𝐶)))
16223oveq1i 7223 . . . . . . 7 (𝑂t (𝐵[,]𝐶)) = ((𝑅t (𝐴[,]𝐶)) ↾t (𝐵[,]𝐶))
163161, 162, 843eqtr4g 2803 . . . . . 6 (𝜑 → (𝑂t (𝐵[,]𝐶)) = 𝑁)
164163, 150oveq12d 7231 . . . . 5 (𝜑 → ((𝑂t (𝐵[,]𝐶)) ×t (𝐽t 𝑋)) = (𝑁 ×t 𝐽))
165159, 164eqtrd 2777 . . . 4 (𝜑 → ((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) = (𝑁 ×t 𝐽))
166165oveq1d 7228 . . 3 (𝜑 → (((𝑂 ×t 𝐽) ↾t ((𝐵[,]𝐶) × 𝑋)) Cn 𝐾) = ((𝑁 ×t 𝐽) Cn 𝐾))
167113, 156, 1663eltr4d 2853 . 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 22191 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 1089   = wceq 1543  wcel 2110  wral 3061  Vcvv 3408  cun 3864  wss 3866  ifcif 4439   cuni 4819   class class class wbr 5053   × cxp 5549  ran crn 5552  cres 5553  wf 6376  cfv 6380  (class class class)co 7213  cmpo 7215  cr 10728  cle 10868  (,)cioo 12935  [,]cicc 12938  t crest 16925  topGenctg 16942  Topctop 21790  TopOnctopon 21807  Clsdccld 21913   Cn ccn 22121   ×t ctx 22457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fi 9027  df-sup 9058  df-inf 9059  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439  df-q 12545  df-ioo 12939  df-icc 12942  df-rest 16927  df-topgen 16948  df-top 21791  df-topon 21808  df-bases 21843  df-cld 21916  df-cn 22124  df-tx 22459
This theorem is referenced by:  htpycc  23877  pcocn  23914  pcohtpylem  23916  pcopt  23919  pcopt2  23920  pcoass  23921  pcorevlem  23923
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