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Mirrors > Home > HSE Home > Th. List > cdj3lem3 | Structured version Visualization version GIF version |
Description: Lemma for cdj3i 32470. Value of the second-component function 𝑇. (Contributed by NM, 23-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdj3lem2.1 | ⊢ 𝐴 ∈ Sℋ |
cdj3lem2.2 | ⊢ 𝐵 ∈ Sℋ |
cdj3lem3.3 | ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))) |
Ref | Expression |
---|---|
cdj3lem3 | ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝐶 +ℎ 𝐷)) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4217 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | eqeq1i 2740 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = 0ℋ ↔ (𝐵 ∩ 𝐴) = 0ℋ) |
3 | cdj3lem2.2 | . . . . . . . 8 ⊢ 𝐵 ∈ Sℋ | |
4 | 3 | sheli 31243 | . . . . . . 7 ⊢ (𝐷 ∈ 𝐵 → 𝐷 ∈ ℋ) |
5 | cdj3lem2.1 | . . . . . . . 8 ⊢ 𝐴 ∈ Sℋ | |
6 | 5 | sheli 31243 | . . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ ℋ) |
7 | ax-hvcom 31030 | . . . . . . 7 ⊢ ((𝐷 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐷 +ℎ 𝐶) = (𝐶 +ℎ 𝐷)) | |
8 | 4, 6, 7 | syl2an 596 | . . . . . 6 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐷 +ℎ 𝐶) = (𝐶 +ℎ 𝐷)) |
9 | 8 | fveq2d 6911 | . . . . 5 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝑇‘(𝐷 +ℎ 𝐶)) = (𝑇‘(𝐶 +ℎ 𝐷))) |
10 | 9 | 3adant3 1131 | . . . 4 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) = 0ℋ) → (𝑇‘(𝐷 +ℎ 𝐶)) = (𝑇‘(𝐶 +ℎ 𝐷))) |
11 | cdj3lem3.3 | . . . . . 6 ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))) | |
12 | 3, 5 | shscomi 31392 | . . . . . . 7 ⊢ (𝐵 +ℋ 𝐴) = (𝐴 +ℋ 𝐵) |
13 | 3 | sheli 31243 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ ℋ) |
14 | 5 | sheli 31243 | . . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ℋ) |
15 | ax-hvcom 31030 | . . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤)) | |
16 | 13, 14, 15 | syl2an 596 | . . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑤 +ℎ 𝑧) = (𝑧 +ℎ 𝑤)) |
17 | 16 | eqeq2d 2746 | . . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (𝑥 = (𝑤 +ℎ 𝑧) ↔ 𝑥 = (𝑧 +ℎ 𝑤))) |
18 | 17 | rexbidva 3175 | . . . . . . . 8 ⊢ (𝑤 ∈ 𝐵 → (∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧) ↔ ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))) |
19 | 18 | riotabiia 7408 | . . . . . . 7 ⊢ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧)) = (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)) |
20 | 12, 19 | mpteq12i 5254 | . . . . . 6 ⊢ (𝑥 ∈ (𝐵 +ℋ 𝐴) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧))) = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤))) |
21 | 11, 20 | eqtr4i 2766 | . . . . 5 ⊢ 𝑇 = (𝑥 ∈ (𝐵 +ℋ 𝐴) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑤 +ℎ 𝑧))) |
22 | 3, 5, 21 | cdj3lem2 32464 | . . . 4 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) = 0ℋ) → (𝑇‘(𝐷 +ℎ 𝐶)) = 𝐷) |
23 | 10, 22 | eqtr3d 2777 | . . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) = 0ℋ) → (𝑇‘(𝐶 +ℎ 𝐷)) = 𝐷) |
24 | 2, 23 | syl3an3b 1404 | . 2 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝐶 +ℎ 𝐷)) = 𝐷) |
25 | 24 | 3com12 1122 | 1 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝐶 +ℎ 𝐷)) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ∩ cin 3962 ↦ cmpt 5231 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 ℋchba 30948 +ℎ cva 30949 Sℋ csh 30957 +ℋ cph 30960 0ℋc0h 30964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-hilex 31028 ax-hfvadd 31029 ax-hvcom 31030 ax-hvass 31031 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvmulass 31036 ax-hvdistr1 31037 ax-hvdistr2 31038 ax-hvmul0 31039 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-grpo 30522 df-ablo 30574 df-hvsub 31000 df-sh 31236 df-ch0 31282 df-shs 31337 |
This theorem is referenced by: cdj3lem3a 32468 cdj3i 32470 |
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