Theorem addsproplem6 28025

Description: Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.)

Hypotheses

Ref	Expression
addsproplem.1	⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
addspropord.2	⊢ (𝜑 → 𝑋 ∈ No )
addspropord.3	⊢ (𝜑 → 𝑌 ∈ No )
addspropord.4	⊢ (𝜑 → 𝑍 ∈ No )
addspropord.5	⊢ (𝜑 → 𝑌 <s 𝑍)
addsproplem6.6	⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))

Assertion

Ref	Expression
addsproplem6	⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem addsproplem6

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addspropord.3	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No )
2		addspropord.4	. . . 4 ⊢ (𝜑 → 𝑍 ∈ No )
3		addsproplem6.6	. . . 4 ⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))
4		addspropord.5	. . . 4 ⊢ (𝜑 → 𝑌 <s 𝑍)
5		nodense 27755	. . . 4 ⊢ (((𝑌 ∈ No ∧ 𝑍 ∈ No ) ∧ (( bday ‘𝑌) = ( bday ‘𝑍) ∧ 𝑌 <s 𝑍)) → ∃𝑚 ∈ No (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
6	1, 2, 3, 4, 5	syl22anc 838	. . 3 ⊢ (𝜑 → ∃𝑚 ∈ No (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
7		addsproplem.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
8		addspropord.2	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ No )
9	7, 8, 1	addsproplem3 28022	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})))
10	9	simp1d 1142	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ No )
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ No )
12	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
13	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑋 ∈ No )
14		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ No )
15		unidm 4180	. . . . . . 7 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑚))) = (( bday ‘𝑋) +no ( bday ‘𝑚))
16		simprr1 1221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑚) ∈ ( bday ‘𝑌))
17		bdayelon 27839	. . . . . . . . . 10 ⊢ ( bday ‘𝑚) ∈ On
18		bdayelon 27839	. . . . . . . . . 10 ⊢ ( bday ‘𝑌) ∈ On
19		bdayelon 27839	. . . . . . . . . 10 ⊢ ( bday ‘𝑋) ∈ On
20		naddel2 8744	. . . . . . . . . 10 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
21	17, 18, 19, 20	mp3an 1461	. . . . . . . . 9 ⊢ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
22	16, 21	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
23		elun1 4205	. . . . . . . 8 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
24	22, 23	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
25	15, 24	eqeltrid 2848	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑚))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
26	12, 13, 14, 14, 25	addsproplem1 28020	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑚) ∈ No ∧ (𝑚 <s 𝑚 → (𝑚 +s 𝑋) <s (𝑚 +s 𝑋))))
27	26	simpld 494	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ No )
28		uncom 4181	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) = ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌)))
29	28	eleq2i 2836	. . . . . . . . . . 11 ⊢ (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) ↔ ((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))))
30	29	imbi1i 349	. . . . . . . . . 10 ⊢ ((((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
31	30	ralbii 3099	. . . . . . . . 9 ⊢ (∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
32	31	2ralbii 3134	. . . . . . . 8 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
33	7, 32	sylib 218	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
34	33, 8, 2	addsproplem3 28022	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑍)𝑔 = (𝑋 +s ℎ)})))
35	34	simp1d 1142	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ No )
36	35	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ No )
37	9	simp3d 1144	. . . . . 6 ⊢ (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
39		ovex 7481	. . . . . . 7 ⊢ (𝑋 +s 𝑌) ∈ V
40	39	snid 4684	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)}
41	40	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)})
42		oldbday 27957	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
43	18, 14, 42	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
44	16, 43	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( O ‘( bday ‘𝑌)))
45		simprr2 1222	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑌 <s 𝑚)
46		breq2 5170	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (𝑌 <s 𝑎 ↔ 𝑌 <s 𝑚))
47		rightval 27921	. . . . . . . . . 10 ⊢ ( R ‘𝑌) = {𝑎 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑌 <s 𝑎}
48	46, 47	elrab2 3711	. . . . . . . . 9 ⊢ (𝑚 ∈ ( R ‘𝑌) ↔ (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑌 <s 𝑚))
49	44, 45, 48	sylanbrc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( R ‘𝑌))
50		eqid 2740	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)
51		oveq2 7456	. . . . . . . . 9 ⊢ (ℎ = 𝑚 → (𝑋 +s ℎ) = (𝑋 +s 𝑚))
52	51	rspceeqv 3658	. . . . . . . 8 ⊢ ((𝑚 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
53	49, 50, 52	sylancl 585	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
54		ovex 7481	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) ∈ V
55		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑔 = (𝑋 +s 𝑚) → (𝑔 = (𝑋 +s ℎ) ↔ (𝑋 +s 𝑚) = (𝑋 +s ℎ)))
56	55	rexbidv 3185	. . . . . . . 8 ⊢ (𝑔 = (𝑋 +s 𝑚) → (∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ) ↔ ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ)))
57	54, 56	elab 3694	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)} ↔ ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
58	53, 57	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})
59		elun2 4206	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)} → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
60	58, 59	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
61	38, 41, 60	ssltsepcd 27857	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑚))
62	33	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
63	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑍 ∈ No )
64	62, 13, 63	addsproplem3 28022	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑍)𝑔 = (𝑋 +s ℎ)})))
65	64	simp2d 1143	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)})
66	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑌) = ( bday ‘𝑍))
67	16, 66	eleqtrd 2846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑚) ∈ ( bday ‘𝑍))
68		bdayelon 27839	. . . . . . . . . . 11 ⊢ ( bday ‘𝑍) ∈ On
69		oldbday 27957	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑍) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑍)))
70	68, 14, 69	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑍)))
71	67, 70	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( O ‘( bday ‘𝑍)))
72		simprr3 1223	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 <s 𝑍)
73		breq1 5169	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (𝑎 <s 𝑍 ↔ 𝑚 <s 𝑍))
74		leftval 27920	. . . . . . . . . 10 ⊢ ( L ‘𝑍) = {𝑎 ∈ ( O ‘( bday ‘𝑍)) ∣ 𝑎 <s 𝑍}
75	73, 74	elrab2 3711	. . . . . . . . 9 ⊢ (𝑚 ∈ ( L ‘𝑍) ↔ (𝑚 ∈ ( O ‘( bday ‘𝑍)) ∧ 𝑚 <s 𝑍))
76	71, 72, 75	sylanbrc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( L ‘𝑍))
77		oveq2 7456	. . . . . . . . 9 ⊢ (𝑑 = 𝑚 → (𝑋 +s 𝑑) = (𝑋 +s 𝑚))
78	77	rspceeqv 3658	. . . . . . . 8 ⊢ ((𝑚 ∈ ( L ‘𝑍) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
79	76, 50, 78	sylancl 585	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
80		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑐 = (𝑋 +s 𝑚) → (𝑐 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
81	80	rexbidv 3185	. . . . . . . 8 ⊢ (𝑐 = (𝑋 +s 𝑚) → (∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
82	54, 81	elab 3694	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
83	79, 82	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)})
84		elun2 4206	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
85	83, 84	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
86		ovex 7481	. . . . . . 7 ⊢ (𝑋 +s 𝑍) ∈ V
87	86	snid 4684	. . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
88	87	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
89	65, 85, 88	ssltsepcd 27857	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) <s (𝑋 +s 𝑍))
90	11, 27, 36, 61, 89	slttrd 27822	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
91	6, 90	rexlimddv 3167	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
92	1, 8	addscomd 28018	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
93	2, 8	addscomd 28018	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
94	91, 92, 93	3brtr4d 5198	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076   ∪ cun 3974  {csn 4648   class class class wbr 5166  Oncon0 6395  ‘cfv 6573  (class class class)co 7448   +no cnadd 8721   No csur 27702   <s cslt 27703   bday cbday 27704   <<s csslt 27843   O cold 27900   L cleft 27902   R cright 27903   +_s cadds 28010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec2 28000  df-adds 28011

This theorem is referenced by:  addsproplem7  28026