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Mirrors > Home > HSE Home > Th. List > cdj3lem2 | Structured version Visualization version GIF version |
Description: Lemma for cdj3i 30221. Value of the first-component function 𝑆. (Contributed by NM, 23-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdj3lem2.1 | ⊢ 𝐴 ∈ Sℋ |
cdj3lem2.2 | ⊢ 𝐵 ∈ Sℋ |
cdj3lem2.3 | ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤))) |
Ref | Expression |
---|---|
cdj3lem2 | ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝐶 +ℎ 𝐷)) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdj3lem2.1 | . . . . 5 ⊢ 𝐴 ∈ Sℋ | |
2 | cdj3lem2.2 | . . . . 5 ⊢ 𝐵 ∈ Sℋ | |
3 | 1, 2 | shsvai 29144 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶 +ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵)) |
4 | eqeq1 2828 | . . . . . . 7 ⊢ (𝑥 = (𝐶 +ℎ 𝐷) → (𝑥 = (𝑧 +ℎ 𝑤) ↔ (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤))) | |
5 | 4 | rexbidv 3300 | . . . . . 6 ⊢ (𝑥 = (𝐶 +ℎ 𝐷) → (∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤) ↔ ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤))) |
6 | 5 | riotabidv 7119 | . . . . 5 ⊢ (𝑥 = (𝐶 +ℎ 𝐷) → (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)) = (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤))) |
7 | cdj3lem2.3 | . . . . 5 ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤))) | |
8 | riotaex 7121 | . . . . 5 ⊢ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6771 | . . . 4 ⊢ ((𝐶 +ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵) → (𝑆‘(𝐶 +ℎ 𝐷)) = (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤))) |
10 | 3, 9 | syl 17 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝑆‘(𝐶 +ℎ 𝐷)) = (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤))) |
11 | 10 | 3adant3 1128 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝐶 +ℎ 𝐷)) = (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤))) |
12 | eqid 2824 | . . . . 5 ⊢ (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝐷) | |
13 | oveq2 7167 | . . . . . 6 ⊢ (𝑤 = 𝐷 → (𝐶 +ℎ 𝑤) = (𝐶 +ℎ 𝐷)) | |
14 | 13 | rspceeqv 3641 | . . . . 5 ⊢ ((𝐷 ∈ 𝐵 ∧ (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝐷)) → ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝑤)) |
15 | 12, 14 | mpan2 689 | . . . 4 ⊢ (𝐷 ∈ 𝐵 → ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝑤)) |
16 | 15 | 3ad2ant2 1130 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝑤)) |
17 | simp1 1132 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → 𝐶 ∈ 𝐴) | |
18 | 1, 2 | cdjreui 30212 | . . . . 5 ⊢ (((𝐶 +ℎ 𝐷) ∈ (𝐴 +ℋ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃!𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤)) |
19 | 3, 18 | stoic3 1776 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃!𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤)) |
20 | oveq1 7166 | . . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧 +ℎ 𝑤) = (𝐶 +ℎ 𝑤)) | |
21 | 20 | eqeq2d 2835 | . . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤) ↔ (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝑤))) |
22 | 21 | rexbidv 3300 | . . . . 5 ⊢ (𝑧 = 𝐶 → (∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤) ↔ ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝑤))) |
23 | 22 | riota2 7142 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃!𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤)) → (∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝑤) ↔ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤)) = 𝐶)) |
24 | 17, 19, 23 | syl2anc 586 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝐶 +ℎ 𝑤) ↔ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤)) = 𝐶)) |
25 | 16, 24 | mpbid 234 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 (𝐶 +ℎ 𝐷) = (𝑧 +ℎ 𝑤)) = 𝐶) |
26 | 11, 25 | eqtrd 2859 | 1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝐶 +ℎ 𝐷)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 ∃!wreu 3143 ∩ cin 3938 ↦ cmpt 5149 ‘cfv 6358 ℩crio 7116 (class class class)co 7159 +ℎ cva 28700 Sℋ csh 28708 +ℋ cph 28711 0ℋc0h 28715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-hilex 28779 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvmulass 28787 ax-hvdistr1 28788 ax-hvdistr2 28789 ax-hvmul0 28790 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-grpo 28273 df-ablo 28325 df-hvsub 28751 df-sh 28987 df-ch0 29033 df-shs 29088 |
This theorem is referenced by: cdj3lem2a 30216 cdj3lem2b 30217 cdj3lem3 30218 cdj3i 30221 |
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