| Step | Hyp | Ref
| Expression |
| 1 | | limccnp2.h |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘〈𝐶, 𝐷〉)) |
| 2 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ ∪ 𝐽 =
∪ 𝐽 |
| 3 | 2 | cnprcl 23253 |
. . . . . . . . . . 11
⊢ (𝐻 ∈ ((𝐽 CnP 𝐾)‘〈𝐶, 𝐷〉) → 〈𝐶, 𝐷〉 ∈ ∪
𝐽) |
| 4 | 1, 3 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ ∪
𝐽) |
| 5 | | limccnp2.j |
. . . . . . . . . . . 12
⊢ 𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) |
| 6 | | limccnp2.k |
. . . . . . . . . . . . . . 15
⊢ 𝐾 =
(TopOpen‘ℂfld) |
| 7 | 6 | cnfldtopon 24803 |
. . . . . . . . . . . . . 14
⊢ 𝐾 ∈
(TopOn‘ℂ) |
| 8 | | txtopon 23599 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ (TopOn‘ℂ)
∧ 𝐾 ∈
(TopOn‘ℂ)) → (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ ×
ℂ))) |
| 9 | 7, 7, 8 | mp2an 692 |
. . . . . . . . . . . . 13
⊢ (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ
× ℂ)) |
| 10 | | limccnp2.x |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑋 ⊆ ℂ) |
| 11 | | limccnp2.y |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑌 ⊆ ℂ) |
| 12 | | xpss12 5700 |
. . . . . . . . . . . . . 14
⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ ×
ℂ)) |
| 13 | 10, 11, 12 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑋 × 𝑌) ⊆ (ℂ ×
ℂ)) |
| 14 | | resttopon 23169 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ
× ℂ)) ∧ (𝑋
× 𝑌) ⊆ (ℂ
× ℂ)) → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌))) |
| 15 | 9, 13, 14 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌))) |
| 16 | 5, 15 | eqeltrid 2845 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐽 ∈ (TopOn‘(𝑋 × 𝑌))) |
| 17 | | toponuni 22920 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ 𝐽) |
| 18 | 16, 17 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝐽) |
| 19 | 4, 18 | eleqtrrd 2844 |
. . . . . . . . 9
⊢ (𝜑 → 〈𝐶, 𝐷〉 ∈ (𝑋 × 𝑌)) |
| 20 | | opelxp 5721 |
. . . . . . . . 9
⊢
(〈𝐶, 𝐷〉 ∈ (𝑋 × 𝑌) ↔ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) |
| 21 | 19, 20 | sylib 218 |
. . . . . . . 8
⊢ (𝜑 → (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) |
| 22 | 21 | simpld 494 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ 𝑋) |
| 23 | 22 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶 ∈ 𝑋) |
| 24 | | simpll 767 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝜑) |
| 25 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵})) |
| 26 | | elun 4153 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵})) |
| 27 | 25, 26 | sylib 218 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵})) |
| 28 | 27 | ord 865 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵})) |
| 29 | | elsni 4643 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) |
| 30 | 28, 29 | syl6 35 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 ∈ 𝐴 → 𝑥 = 𝐵)) |
| 31 | 30 | con1d 145 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 = 𝐵 → 𝑥 ∈ 𝐴)) |
| 32 | 31 | imp 406 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴) |
| 33 | | limccnp2.r |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋) |
| 34 | 24, 32, 33 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑅 ∈ 𝑋) |
| 35 | 23, 34 | ifclda 4561 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, 𝑅) ∈ 𝑋) |
| 36 | 21 | simprd 495 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈ 𝑌) |
| 37 | 36 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐷 ∈ 𝑌) |
| 38 | | limccnp2.s |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌) |
| 39 | 24, 32, 38 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑆 ∈ 𝑌) |
| 40 | 37, 39 | ifclda 4561 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐷, 𝑆) ∈ 𝑌) |
| 41 | 35, 40 | opelxpd 5724 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉 ∈ (𝑋 × 𝑌)) |
| 42 | | eqidd 2738 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉)) |
| 43 | 7 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈
(TopOn‘ℂ)) |
| 44 | | cnpf2 23258 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘〈𝐶, 𝐷〉)) → 𝐻:(𝑋 × 𝑌)⟶ℂ) |
| 45 | 16, 43, 1, 44 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → 𝐻:(𝑋 × 𝑌)⟶ℂ) |
| 46 | 45 | feqmptd 6977 |
. . . 4
⊢ (𝜑 → 𝐻 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝐻‘𝑦))) |
| 47 | | fveq2 6906 |
. . . . 5
⊢ (𝑦 = 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉 → (𝐻‘𝑦) = (𝐻‘〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉)) |
| 48 | | df-ov 7434 |
. . . . . 6
⊢ (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = (𝐻‘〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) |
| 49 | | ovif12 7533 |
. . . . . 6
⊢ (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆)) |
| 50 | 48, 49 | eqtr3i 2767 |
. . . . 5
⊢ (𝐻‘〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆)) |
| 51 | 47, 50 | eqtrdi 2793 |
. . . 4
⊢ (𝑦 = 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉 → (𝐻‘𝑦) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) |
| 52 | 41, 42, 46, 51 | fmptco 7149 |
. . 3
⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉)) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆)))) |
| 53 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅) |
| 54 | 53, 33 | dmmptd 6713 |
. . . . . . . . . 10
⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝑅) = 𝐴) |
| 55 | | limccnp2.c |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵)) |
| 56 | | limcrcl 25909 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑅):dom (𝑥 ∈ 𝐴 ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
| 57 | 55, 56 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑅):dom (𝑥 ∈ 𝐴 ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
| 58 | 57 | simp2d 1144 |
. . . . . . . . . 10
⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ) |
| 59 | 54, 58 | eqsstrrd 4019 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ⊆ ℂ) |
| 60 | 57 | simp3d 1145 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 61 | 60 | snssd 4809 |
. . . . . . . . 9
⊢ (𝜑 → {𝐵} ⊆ ℂ) |
| 62 | 59, 61 | unssd 4192 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ) |
| 63 | | resttopon 23169 |
. . . . . . . 8
⊢ ((𝐾 ∈ (TopOn‘ℂ)
∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) →
(𝐾 ↾t
(𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵}))) |
| 64 | 7, 62, 63 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → (𝐾 ↾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵}))) |
| 65 | | ssun2 4179 |
. . . . . . . 8
⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵}) |
| 66 | | snssg 4783 |
. . . . . . . . 9
⊢ (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵}))) |
| 67 | 60, 66 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵}))) |
| 68 | 65, 67 | mpbiri 258 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ (𝐴 ∪ {𝐵})) |
| 69 | 10 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑋 ⊆ ℂ) |
| 70 | 69, 33 | sseldd 3984 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ ℂ) |
| 71 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝐾 ↾t (𝐴 ∪ {𝐵})) = (𝐾 ↾t (𝐴 ∪ {𝐵})) |
| 72 | 59, 60, 70, 71, 6 | limcmpt 25918 |
. . . . . . . 8
⊢ (𝜑 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))) |
| 73 | 55, 72 | mpbid 232 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)) |
| 74 | | limccnp2.d |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵)) |
| 75 | 11 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ⊆ ℂ) |
| 76 | 75, 38 | sseldd 3984 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℂ) |
| 77 | 59, 60, 76, 71, 6 | limcmpt 25918 |
. . . . . . . 8
⊢ (𝜑 → (𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))) |
| 78 | 74, 77 | mpbid 232 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)) |
| 79 | 64, 43, 43, 68, 73, 78 | txcnp 23628 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP (𝐾 ×t 𝐾))‘𝐵)) |
| 80 | 9 | topontopi 22921 |
. . . . . . . 8
⊢ (𝐾 ×t 𝐾) ∈ Top |
| 81 | 80 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝐾 ×t 𝐾) ∈ Top) |
| 82 | 41 | fmpttd 7135 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌)) |
| 83 | | toponuni 22920 |
. . . . . . . . . 10
⊢ ((𝐾 ↾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾t (𝐴 ∪ {𝐵}))) |
| 84 | 64, 83 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾t (𝐴 ∪ {𝐵}))) |
| 85 | 84 | feq2d 6722 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉):∪ (𝐾 ↾t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌))) |
| 86 | 82, 85 | mpbid 232 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉):∪ (𝐾 ↾t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌)) |
| 87 | | eqid 2737 |
. . . . . . . 8
⊢ ∪ (𝐾
↾t (𝐴
∪ {𝐵})) = ∪ (𝐾
↾t (𝐴
∪ {𝐵})) |
| 88 | 9 | toponunii 22922 |
. . . . . . . 8
⊢ (ℂ
× ℂ) = ∪ (𝐾 ×t 𝐾) |
| 89 | 87, 88 | cnprest2 23298 |
. . . . . . 7
⊢ (((𝐾 ×t 𝐾) ∈ Top ∧ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉):∪ (𝐾 ↾t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) →
((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP (𝐾 ×t 𝐾))‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵))) |
| 90 | 81, 86, 13, 89 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP (𝐾 ×t 𝐾))‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵))) |
| 91 | 79, 90 | mpbid 232 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵)) |
| 92 | 5 | oveq2i 7442 |
. . . . . 6
⊢ ((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))) |
| 93 | 92 | fveq1i 6907 |
. . . . 5
⊢ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵) |
| 94 | 91, 93 | eleqtrrdi 2852 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵)) |
| 95 | | iftrue 4531 |
. . . . . . . . 9
⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, 𝑅) = 𝐶) |
| 96 | | iftrue 4531 |
. . . . . . . . 9
⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐷, 𝑆) = 𝐷) |
| 97 | 95, 96 | opeq12d 4881 |
. . . . . . . 8
⊢ (𝑥 = 𝐵 → 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉 = 〈𝐶, 𝐷〉) |
| 98 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) |
| 99 | | opex 5469 |
. . . . . . . 8
⊢
〈𝐶, 𝐷〉 ∈ V |
| 100 | 97, 98, 99 | fvmpt 7016 |
. . . . . . 7
⊢ (𝐵 ∈ (𝐴 ∪ {𝐵}) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉)‘𝐵) = 〈𝐶, 𝐷〉) |
| 101 | 68, 100 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉)‘𝐵) = 〈𝐶, 𝐷〉) |
| 102 | 101 | fveq2d 6910 |
. . . . 5
⊢ (𝜑 → ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉)‘𝐵)) = ((𝐽 CnP 𝐾)‘〈𝐶, 𝐷〉)) |
| 103 | 1, 102 | eleqtrrd 2844 |
. . . 4
⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉)‘𝐵))) |
| 104 | | cnpco 23275 |
. . . 4
⊢ (((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉)‘𝐵))) → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)) |
| 105 | 94, 103, 104 | syl2anc 584 |
. . 3
⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ 〈if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)〉)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)) |
| 106 | 52, 105 | eqeltrrd 2842 |
. 2
⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)) |
| 107 | 45 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻:(𝑋 × 𝑌)⟶ℂ) |
| 108 | 107, 33, 38 | fovcdmd 7605 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅𝐻𝑆) ∈ ℂ) |
| 109 | 59, 60, 108, 71, 6 | limcmpt 25918 |
. 2
⊢ (𝜑 → ((𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) limℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))) |
| 110 | 106, 109 | mpbird 257 |
1
⊢ (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) limℂ 𝐵)) |